Chapter 7 confidence intervals and sample size answers

Chapter 7

Estimation

Chapter Learning Objectives

1. Explain the concepts of estimation, point estimates, confidence level, and confidence interval 2. Calculate and interpret confidence intervals for means 3. Describe the concept of risk and how to reduce it 4. Calculate and interpret confidence intervals for proportions

In this chapter, we discuss the procedures involved in estimating population means and proportions based on the principles of sampling and statistical inference discussed in Chapter 6 (“Sampling and Sampling Distributions”). Knowledge about the sampling distribution allows us to estimate population means and proportions from sample outcomes and to assess the accuracy of these estimates. Consider three examples of information derived from samples.

Example 1: Based on a random sample of 1,019 U.S. adults, a March 2016 Gallup poll found that the percentage of Americans who identify as environmentalists has decreased. Compared with the 1991 high of 78%, in 2016 only 42% of Americans self-identified as environmentalists. In its report, Gallup attributed the decline to several factors, including the adoption of routine environmental friendly practices and the politicization of environmental issues.1

Example 2: Every other year, the National Opinion Research Center conducts the General Social Survey (GSS) on a representative sample of about 1,500 respondents. The GSS, from which many of the examples in this book are selected, is designed to provide social science researchers with a readily accessible database of socially relevant attitudes, behaviors, and attributes of a cross-section of the U.S. adult population. For example, in analyzing the responses to the 2014 GSS, researchers found that the average respondent’s education was about 13.77 years. This average probably differs from the average of the population from which the GSS sample was drawn. However, we can establish that in most cases the sample mean (in this case, 13.77 years) is fairly close to the actual true average in the population.

Example 3: In 2016, North Carolina legislators passed House Bill 2, prohibiting transgender people from using bathrooms and locker rooms that do not match the gender on their birth certificate. The law quickly drew protests from civil rights and

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LGBT (lesbian, gay, bisexual, transgender) rights groups. A CNN/ORC poll of 1,001 Americans revealed that 39% of those surveyed strongly oppose laws that require transgender individuals to use restroom facilities that correspond on their gender at birth rather than their gender identity. Seventy-five percent favor laws guaranteeing equal protection for transgender people in jobs, housing, and public accomodations.2

The percentage of Americans who identify as environmentalists as approximated by the Gallup organization, the average level of education in the United States as calculated from the GSS, and the percentage of Americans who oppose laws requiring transgender people to use bathrooms that correspond to their gender at birth based on the CNN/ORC poll are all sample estimates of population parameters. Population parameters are the actual percentage of Americans who identify as environmentalists or the actual average level of education in the United States. The 39% who oppose laws that require transgender individuals to use restrooms that correspond to their birth gender can be used to estimate the actual percentage of all Americans who oppose such laws.

These are illustrations of estimation. Estimation is a process whereby we select a random sample from a population and use a sample statistic to estimate a population parameter. We can use sample proportions as estimates of population proportions, sample means as estimates of population means, or sample variances as estimates of population variances.

Why estimate? The goal of most research is to find the population parameter. Yet we hardly ever have enough resources to collect information about the entire population. We rarely know the actual value of the population parameter. On the other hand, we can learn a lot about a population by randomly selecting a sample from that population and obtaining an estimate of the population parameter. The major objective of sampling theory and statistical inference is to provide estimates of unknown population parameters from sample statistics.

Estimation A process whereby we select a random sample from a population and use a sample statistic to estimate a population parameter.

Point and Interval Estimation

Estimates of population characteristics can be divided into two types: (1) point estimates and (2) interval estimates. Point estimates are sample statistics used to estimate the exact value of a population parameter. When the Gallup organization reports that 42% of Americans identify as environmentalists, they are using a point estimate. Similarly, if we reported the average level of education of the population of adult Americans to be exactly 13.77 years, we would be using a point estimate.

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Point estimate A sample statistic used to estimate the exact value of a population parameter.

Confidence interval (CI) A range of values defined by the confidence level within which the population parameter is estimated to fall. A confidence interval may also be referred to as a margin of error.

Confidence level The likelihood, expressed as a percentage or a probability, that a specified interval will contain the population parameter.

The problem with point estimates is that sample statistics vary, usually resulting in some sort of sampling error. Thus, we never really know how accurate they are. As a result we rarely rely on them as estimators of population parameters such as average income or percentage of the population who are in favor of gay and lesbian relations.

One method of increasing accuracy is to use an interval estimate rather than a point estimate. In interval estimation, we identify a range of values within which the population parameter may fall. This range of values is called a confidence interval (CI). Instead of using a single value, 13.77 years, as an estimate of the mean education of adult Americans, we could say that the population mean is somewhere between 12 and 14 years.

When we use confidence intervals to estimate population parameters, such as mean educational levels, we can also evaluate the accuracy of this estimate by assessing the likelihood that any given interval will contain the mean. This likelihood, expressed as a percentage or a probability, is called a confidence level. Confidence intervals are defined in terms of confidence levels. Thus, by selecting a 95% confidence level, we are saying that there is a .95 probability—or 95 chances out of 100—that a specified interval will contain the population mean. Confidence intervals can be constructed for any level of confidence, but the most common ones are the 90%, 95%, and 99% levels. You should also know that confidence intervals are sometimes referred to in terms of margin of error. In short, margin of error is simply the radius of a confidence interval. If we select a 95% confidence level, we would have a 5% chance of our interval being incorrect.

Confidence intervals can be constructed for many different parameters based on their corresponding sample statistics. In this chapter, we describe the rationale and the procedure for the construction of confidence intervals for means and proportions.

Margin of error The radius of a confidence interval.

Learning Check 7.1

What is the difference between a point estimate and a confidence interval?

Confidence Intervals for Means

To illustrate the procedure for establishing confidence intervals for means, we’ll reintroduce one of the research examples mentioned in Chapter 6—assessing the needs of commuter students.

Recall that we have been given enough money to survey a random sample of 500 students. One of our tasks is to estimate the average commuting time of all 15,000 commuters on our campus—the population parameter. To obtain this estimate, we calculate the average commuting time for the sample. Suppose the sample average is Ῡ = 7.5 hours/week, and we want to use it as an estimate of the true average commuting time for the entire population of commuting students.

Because it is based on a sample, this estimate is subject to sampling error. We do not know how close it is to the true population mean. However, based on what the central limit theorem tells us about the properties of the sampling distribution of the mean, we know that with a large enough sample size, most sample means will tend to be close to the true population mean. Therefore, it is unlikely that our sample mean, Ῡ = 7.5 hours/week, deviates much from the true population mean.

We know that the sampling distribution of the mean is approximately normal with a mean equal to the population mean µ and a standard error σῩ (standard deviation of the sampling distribution) as follows:

(7.1)

σ√N

This information allows us to use the normal distribution to determine the probability that a sample mean will fall within a certain distance—measured in standard deviation (standard error) units or Z scores—of µ or µῩ We can make the following assumptions:

• A total of 68% of all random sample means will fall within ±1 standard error of the true population mean.

• A total of 95% of all random sample means will fall within ±1.96 standard errors of the true population mean.

• A total of 99% of all random sample means will fall within ±2.58 standard errors of the true population mean.

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A Closer Look 7.1

Estimation as a Type of Inference

Using inferential statistics, a researcher is able to describe a population based entirely on information from a sample of that population. A confidence interval is an example of this—by knowing a sample mean, sample size, and sample standard deviation, we are able to say something about the population from which that sample was drawn.

Combining this information gives us a range within which we can confidently say that the population mean falls.

On the basis of these assumptions and the value of the standard error, we can establish a range of values—a confidence interval—that is likely to contain the actual population mean. We can also evaluate the accuracy of this estimate by assessing the likelihood that this range of values will actually contain the population mean.

The general formula for constructing a confidence interval (CI) for any level is

(7.2)

CI=¯¯̄ Y±Z(σ¯¯̄Y)CI=Y¯±Z(σY¯)

Note that to calculate a confidence interval, we take the sample mean and add to or subtract from it the product of a Z value and the standard error.

The Z score we choose depends on the desired confidence level. For example, to obtain a 95% confidence interval we would choose a Z of 1.96 because we know (from Appendix B) that 95% of the area under the curve lies between ±1.96. Similarly, for a 99% confidence level, we would choose a Z of 2.58. The relationship between the confidence level and Z is illustrated in Figure 7.1 for the 95% and 99% confidence levels.

Learning Check 7.2

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To understand the relationship between the confidence level and Z, review the material in Chapter 5. What would be the corresponding Z value for a 98% confidence interval?

Determining the Confidence Interval

To determine the confidence interval for means, follow these steps:

1. Calculate the standard error of the mean. 2. Decide on the level of confidence, and find the corresponding Z value. 3. Calculate the confidence interval. 4. Interpret the results.

Let’s return to the problem of estimating the mean commuting time of the population of students on our campus. How would you find the 95% confidence interval?

Calculating the Standard Error of the Mean

Let’s suppose that the standard deviation for our population of commuters is σ = 1.5. We calculate the standard error for the sampling distribution of the mean:

σ¯¯̄Y=σ√N=1.5√500=0.07

Deciding on the Level of Confidence and Finding the Corresponding Z Value

We decide on a 95% confidence level. The Z value corresponding to a 95% confidence level is 1.96.

Calculating the Confidence Interval

The confidence interval is calculated by adding and subtracting from the observed sample mean the product of the standard error and Z:

95% CI = 7.5 ± 1.96(0.07)

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= 7.5 ± 0.14

= 7.36 to 7.64

The 95% CI for the mean commuting time is illustrated in Figure 7.2.

Figure 7.1 Relationship Between Confidence Level and Z for 95% and 99% Confidence Intervals

Source: David Freedman, Robert Pisani, Roger Purves, and Ani Akhikari, Statistics, 2nd ed. (New York: Norton, 1991).

Figure 7.2 Ninety-Five Percent Confidence Interval for the Mean Commuting Time (N = 500)

Interpreting the Results

We can be 95% confident that the actual mean commuting time—the true population mean—is not less than 7.36 hours and not greater than 7.64 hours. In other words, if we collected a large number of samples (N = 500) from the population of commuting students, 95 times out of 100, the true population mean would be included within our computed interval. With a 95% confidence level, there is a 5% risk that we are wrong. Five times out of 100, the true population mean will not be included in the specified interval.

Remember that we can never be sure whether the population mean is actually contained within the confidence interval. Once the sample is selected and the confidence interval defined, the confidence interval either does or does not contain the population mean—but we will never be sure.

Learning Check 7.3

What is the 90% confidence interval for the mean commuting time? First, find the Z value associated with a 90% confidence level.

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To further illustrate the concept of confidence intervals, let’s suppose that we draw 10 different samples (N = 500) from the population of commuting students. For each sample mean, we construct a 95% confidence interval. Figure 7.3 displays these confidence intervals. Each horizontal line represents a 95% confidence interval constructed around a sample mean (marked with a circle).

The vertical line represents the population mean. Note that the horizontal lines that intersect the vertical line are the intervals that contain the true population mean. Only 1 out of the 10 confidence intervals does not intersect the vertical line, meaning it does not contain the population mean. What would happen if we continued to draw samples of the same size from this population and constructed a 95% confidence interval for each sample? For about 95% of all samples, the specified interval would contain the true population mean, but for 5% of all samples it would not.

Reducing Risk

One way to reduce the risk of being incorrect is by increasing the level of confidence. For instance, we can increase our confidence level from 95% to 99%. The 99% confidence interval for our commuting example is

Figure 7.3 Ninety-Five Percent Confidence Intervals for 10 Samples

99% CI = 7.5 ± 2.58(0.07)

= 7.5 ± 0.18

= 7.32 to 7.68

When using the 99% confidence interval, there is only a 1% risk that we are wrong and the specified interval does not contain the true population mean. We can be almost certain that the true population mean is included in the interval ranging from 7.32 to 7.68 hours/week. Note that by increasing the confidence level, we have also increased the width of the confidence interval from 0.28 (7.36–7.64) to 0.36 hours (7.32–7.68), thereby making our estimate less precise.

This is the trade-off between achieving greater confidence in an estimate and the precision of that estimate. Although using a higher level of confidence increases our confidence that the true population mean is included in our confidence interval, the estimate becomes less precise as the width of the interval increases. Although we are only 95% confident that the interval ranging between 7.36 and 7.64 hours includes the true population mean, it is a more precise estimate

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than the 99% interval ranging from 7.32 to 7.68 hours. The relationship between the confidence level and the precision of the confidence interval is illustrated in Figure 7.4. Table 7.1 lists three commonly used confidence levels along with their corresponding Z values.

Table 7.1 Confidence Levels and Corresponding Z Values

Confidence Level Z Value

90% 1.65

95% 1.96

99% 2.58

Figure 7.4 Confidence Intervals, 95% Versus 99% (Mean Commuting Time)

Estimating Sigma

To calculate confidence intervals, we need to know the standard error of the sampling distribution. The standard error is a function of the population standard deviation and the sample size:

σ√N

In our commuting example, we have been using a hypothetical value, σ = 1.5, for the population standard deviation. Typically, both the mean (µ) and the standard deviation (σ) of the population are unknown to us. When N ≥ 50, however, the sample standard deviation, s, is a good estimate of σῩ The standard error is then calculated as follows:

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(7.3)

s√N

As an example, we’ll estimate the mean hours per day that Americans spend watching television based on the 2014 GSS. The mean hours per day spent watching television for a sample of N = 1,001 is Ῡ = 2.94, and the standard deviation s = 2.60 hours. Let’s determine the 95% confidence interval for these data.

Calculating the Estimated Standard Error of the Mean

The estimated standard error for the sampling distribution of the mean is

2.60√1,001

Deciding on the Level of Confidence and Finding the Corresponding Z Value

We decide on a 95% confidence level. The Z value corresponding to a 95% confidence level is 1.96.

Calculating the Confidence Interval

The confidence interval is calculated by adding to and subtracting from the observed sample mean the product of the standard error and Z:

95% CI = 2.94 ± 1.96(0.08)

= 2.94 ± 0.16

= 2.78 to 3.10

Interpreting the Results

We can be 95% confident that the actual mean hours spent watching television by Americans from which the GSS sample was taken is not less than 2.78 hours and not greater than 3.10 hours. In other words, if we drew a large number of

samples (N = 1,001) from this population, then 95 times out of 100, the true population mean would be included within our computed interval.

Sample Size and Confidence Intervals

Researchers can increase the precision of their estimate by increasing the sample size. In Chapter 6, we learned that larger samples result in smaller standard errors and, therefore, sampling distributions are more clustered around the population mean (Figure 6.6). A more tightly clustered sampling distribution means that our confidence intervals will be narrower and more precise. To illustrate the relationship between sample size and the standard error, and thus the confidence interval, let’s calculate the 95% confidence interval for our GSS data with (a) a sample of N = 150 and (b) a sample of N = 1,987.

With a sample size N = 150, the estimated standard error for the sampling distribution is

s√N

and the 95% confidence interval is

95% CI = 2.94 ± 1.96(0.21)

= 2.94 ± 0.41

= 2.53 to 3.35

With a sample size N = 1,987, the estimated standard error for the sampling distribution is

s√N

and the 95% confidence interval is

95% CI = 2.94 ± 1.96(0.06)

= 2.94 ± 0.12

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= 2.82 to 3.06

In Table 7.2, we summarize the 95% confidence intervals for the mean number of hours watching television for these three sample sizes: N = 150, N = 1,001, and N = 1,987.

Table 7.2 Ninety-Five Percent Confidence Interval and Width for Mean Number of Hours per Day Watching Television for Three

Different Sample Sizes

Sample Size (N) Confidence Interval Interval Width s sῩ

150 2.53–3.35 0.82 2.60 0.21

1,001 2.78–3.10 0.32 2.60 0.08

1,987 2.82–3.06 0.24 2.60 0.12

Note that there is an inverse relationship between sample size and the width of the confidence interval. The increase in sample size is linked with increased precision of the confidence interval. The 95% confidence interval for the GSS sample of 150 cases is 0.82 hours. But the interval widths decrease to 0.32 and 0.24 hours, respectively, as the sample sizes increase to N = 1,001 and then to N = 1,987. We had to nearly double the size of the sample (from 1,001 to 1,987) to reduce the confidence interval by about one fourth (from 0.32 to 0.24 hours). In general, although the precision of estimates increases steadily with sample size, the gains would appear to be rather modest after N reaches 1,987. An important factor to keep in mind is the increased cost associated with a larger sample. Researchers have to consider at what point the increase in precision is too small to justify the additional cost associated with a larger sample.

Learning Check 7.4

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Why do smaller sample sizes produce wider confidence intervals? (See Figure 7.5.) Compare the standard errors of the mean for the three sample sizes.

Statistics in Practice: Hispanic Migration and Earnings

There were nearly 55 million people of Hispanic or Latino ethnicity in the United States in 2014. Mexican-origin Hispanics comprise the single largest group of Hispanics in the country. But the origins of the Hispanic population have diversified with the addition of immigrants from other Latin American nations and Puerto Rico.3

During the past few decades, numerous studies have noted the discrepancy in earnings between these groups.4 The gap in earnings has been attributed mainly to differences in migration status and in level of education. In this classic study, Marta Tienda and Franklin Wilson argued that Mexicans, Puerto Ricans, and Cubans varied markedly in socioeconomic characteristics because of differences in the timing and circumstances of their immigration to the United States.5 The period of entry and the circumstances prompting migration affected the geographical distribution and the employment opportunities of each group. For example, Puerto Ricans were disproportionately located in the Northeast, where the labor market was characterized by the highest unemployment rates, whereas the majority of Cuban immigrants resided in the Southeast, where the unemployment rate was the lowest in the United States.

Tienda and Wilson also noted persistent differences in educational levels among Mexicans and Puerto Ricans compared with Cubans.6 Only about 9% of Mexicans and 16% of Puerto Ricans have graduated college, compared with 25% of Cuban men.7 These differences in migrant status and educational level were likely to be reflected in disparities in earnings among the three groups. We would anticipate that the earnings of Cubans would be higher than the earnings of Mexicans and Puerto Ricans.

A Closer Look 7.2

What Affects Confidence Interval Width? Summary

“Holding other factors constant. . . ” ↑ the confidence interval becomes more precise. → ←

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If the sample size goes up

If the level of confidence goes down (from 99% to 95%) ↓ the confidence interval becomes less precise

.

← →

If the sample size goes down ↑ the confidence interval becomes less precise. ← →

If the value of the sample standard deviation goes up ↓ the confidence interval becomes more precise. → ←

If the value of the sample standard deviation goes down ↑ the confidence interval becomes less precise. ← →

If the level of confidence goes up (from 95% to 99%) ↓ the confidence interval becomes more precise → ←

We tested the ideas of Tienda and Wilson based on a sample of men from the 2000 Census that included 29,233 Cubans, 34,620 Mexican Americans, and 66,933 Puerto Ricans. As hypothesized, with average earnings of $24,018 (s = $36,298), Cubans were at the top of the income hierarchy. Puerto Ricans were intermediate among the groups with earnings averaging $18,748 (s = $25,694). Mexican men were at the bottom of the income hierarchy with average annual earnings of $16,537 (s = $23,502). Although Tienda and Wilson did not calculate confidence intervals for their estimates, we use the data from the 2000 Census to calculate a 95% confidence interval for the mean income of the three groups of Hispanic men.8

To find the 95% confidence interval for Cuban income, we first estimate the standard error:

36,298√29,233

Then, we calculate the confidence interval:

95% CI = 24,018 ± 1.96(212.30)

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= 24,018 ± 416

= 23,602 to 24,434

For Puerto Rican income, the estimated standard error is

25,694√66,933

Figure 7.5 The Relationship Between Sample Size and Confidence Interval Width

and the 95% confidence interval is

95% CI = 18,748 ± 1.96(99.31)

= 18,748 ± 195

= 18,553 to 18,943

Finally, for Mexican income, the estimated standard error is

23,502√34,620

Figure 7.6 Ninety-Five Percent Confidence Intervals for the Mean Income of Puerto Ricans, Mexicans, and Cubans

and the 95% confidence interval is

95% CI = 16,537 ± 1.96(126.31)

= 16,537 ± 248

= 16,289 to 16,785

The confidence intervals for mean annual income of Cuban, Puerto Rican, and Mexican immigrants are illustrated in Figure 7.6. We can say with 95% confidence that the true income mean for each Hispanic group lies somewhere within the corresponding confidence interval. Note that the confidence intervals do not overlap, thus revealing great disparities in earnings among the three groups. Highest interval estimates are for Cubans, followed by Mexicans and then Puerto Ricans.

Confidence Intervals for Proportions

You may already be familiar with confidence intervals, having seen them applied in opinion and election polls like those conducted by Gallup or CNN/ORC. Pollsters interview a random sample representative of a defined population to assess their opinion on a certain issue or their voting preference. Sample proportions or percentages are usually reported along with a margin error, plus or minus a particular value. The margin of error is the confidence interval and is used to estimate the population proportions or percentages. The same conceptual foundations of sampling and statistical inference that are central to the estimation of population means—the selection of random samples and the special properties of the sampling distribution—are also central to the estimation of population proportions.

Earlier, we saw that the sampling distribution of the means underlies the process of estimating population means from sample means. Similarly, the sampling distribution of proportions underlies the estimation of population proportions from sample proportions. Based on the central limit theorem, we know that with sufficient sample size the sampling distribution of proportions is approximately normal, with mean µpequal to the population proportion π and with a standard error of proportions (the standard deviation of the sampling distribution of proportions) equal to

(7.4)

(π)(1−π)N

where

• σp = the standard error of proportions

• π = the population proportion

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• N = the population size

However, since the population proportion, π, is unknown to us (that is what we are trying to estimate), we can use the sample proportion, p, as an estimate of π. The estimated standard error then becomes

(7.5)

(p)(1−p)N

where

• sp = the estimated standard error of proportions

• p = the sample proportion

• N = the sample size

Let’s calculate the estimated standard error for the 2016 Gallup survey about environmentalist identification. Based on a random sample of 1,019 adults, the percentage who identify as an environmentalist was reported at 42%. Based on Formula 7.5, with p = 0.42, 1 − p = (1 − 0.42) = 0.58, and N = 1,019, the standard error is sp=√(.42(1−.42)/1,019=0.015=0.02sp=(.42(1−.42)/1,019=0.015=0.02 . We will have to consider two factors to meet the assumption of normality with the sampling distribution of proportions: (1) the sample size N and (2) the sample proportions p and 1 − p. When p and 1 − p are about 0.50, a sample size of at least 50 is sufficient. But when p > 0.50 (or 1 − p < 0.50), a larger sample is required to meet the assumption of normality. Usually, a sample of 100 or more is adequate for any single estimate of a population proportion.

Determining the Confidence Interval

Because the sampling distribution of proportions is approximately normal, we can use the normal distribution to establish confidence intervals for proportions in the same manner that we used the normal distribution to establish confidence intervals or means.

The general formula for constructing confidence intervals for proportions for any level of confidence is

(7.6)

CI=p±Z(sp)CI=p±Z(sp)

where

• CI = the confidence interval

• p = the observed sample proportion

• Z = the Z corresponding to the confidence level

• sp = the estimated standard error of proportions

Let’s examine this formula in more detail. Note that to obtain a confidence interval at a certain level, we take the sample proportion and add to or subtract from it the product of a Z value and the standard error. The Z value we choose depends on the desired confidence level. We want the area between the mean and the selected ±Z to be equal to the confidence level.

For example, to obtain a 95% confidence interval, we would choose a Z of 1.96 because we know (from Appendix B) that 95% of the area under the curve is included between ±1.96. Similarly, for a 99% confidence level, we would choose a Z of 2.58. (The relationship between confidence level and Z values is illustrated in Figure 7.1.)

To determine the confidence interval for a proportion, we follow the same steps that were used to find confidence intervals for means:

1. Calculate the estimated standard error of the proportion. 2. Decide on the desired level of confidence, and find the corresponding Z value. 3. Calculate the confidence interval. 4. Interpret the results.

To illustrate these steps, we use the Gallup survey results about the percentage of Americans who identify as environmentalists.

Calculating the Estimated Standard Error of the Proportion

The standard error of the proportion 0.42 (42%) with a sample N = 1,019 is 0.02.

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Deciding on the Desired Level of Confidence and Finding the Corresponding Z Value

We choose the 95% confidence level. The Z corresponding to a 95% confidence level is 1.96.

Calculating the Confidence Interval

We calculate the confidence interval by adding to and subtracting from the observed sample proportion the product of the standard error and Z:

95% CI = 0.42 ± 1.96(0.02)

= 0.42 ± 0.04

= 0.38 to 0.46

Interpreting the Results

We are 95% confident that the true population proportion is somewhere between 0.38 and 0.46. In other words, if we drew a large number of samples from the population of adults, then 95 times out of 100, the confidence interval we obtained would contain the true population proportion. We can also express this result in percentages and say that we are 95% confident that the true population percentage of Americans who identify as environmentalists is between 38% and 46%.

Learning Check 7.5

Calculate the 95% confidence interval for the CNN/ORC survey results for those who do not support anti-trans bathroom legislation.

Note that with a 95% confidence level, there is a 5% risk that we are wrong. If we continued to draw large samples from this population, in 5 out of 100 samples the true population proportion would not be included in the specified interval.

We can decrease our risk by increasing the confidence level from 95% to 99%.

99% CI = 0.42 ± 2.58(0.02)

= 0.42 ± 0.05

= .37 to .47

When using the 99% confidence interval, we can be almost certain (99 times out of 100) that the true population proportion is included in the interval ranging from .37 to .47 (or 37% to 47%). However, as we saw earlier, there is a trade- off between achieving greater confidence in making an estimate and the precision of that estimate. Although using a 99% level increased our confidence level from 95% to 99% (thereby reducing our risk of being wrong from 5% to 1%), the estimate became less precise as the width of the interval increased.9

Reading the Research Literature: Women Victims of Intimate

Violence

Janet Fanslow and Elizabeth Robinson (2010)10 studied help-seeking behavior and motivation among women victims of intimate partner violence in New Zealand. The researchers argue that

historically, responses to victims have been developed on the basis of identified need and active advocacy work rather than driven by data. . . .Questions remain, however, in terms of whether the responses that have been developed are the most appropriate ways to deliver help to victims of intimate partner violence according to victims’ personal perceptions of who they would like to receive help from and the types and nature of help they would like to receive. (p. 930)

Fanslow and Robinson relied on data from the New Zealand Violence Against Women Study to document the reasons why victims sought help or left their partner due to domestic violence.

We present one of the tables featured in Fanslow and Robinson’s study (see Table 7.3). Note that 95% confidence intervals are reported for each reason category. The confidence intervals are based on the confidence level, the standard error of the proportion (which can be estimated from p), and the sample size. For example, we know that based on their sample of 486 women, 48.5% reported that “could not endure more” was a reason for seeking help. Based on the 95% confidence interval, we can say that the actual proportion of women victims who identify that they “could not endure more” as a reason for seeking help lies somewhere between 43.6% and 53.3%.

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Fanslow and Robinson identify the categories with the highest percentages for each sample: for seeking help (could not endure more, encouraged by friends or family, children suffering and badly injured) and for leaving (could not endure more, he threatened to kill her, children suffering and badly injured). They write as follows:

The majority of women who sought help, and the majority of women who left violent relationships reported doing so because they “could not endure more”. Where and when this line is drawn will undoubtedly differ from person to person . . . but factors that may affect it include realistic assessment of the man’s behavior and his likelihood to change, and recognition of the seriousness of the violence they experienced. Other commonly reported reasons for leaving and/or seeking help included because women experienced serious injury or feared for their lives. This emphasizes that the violence experienced by many women in the context of intimate relationships is not trivial. (p. 946)11

Concern for children suffering was also identified as an important reason for female victims to seek help and/or to leave their abuser.

When estimates are reported for subgroups, the confidence intervals are likely to vary. Even when a confidence interval is reported only for the overall sample, we can easily compute separate confidence intervals for each of the subgroups if the confidence level and the size of each of the subgroups are included.

Table 7.3 Percentage of Women With Lifetime Experience of Physical and/or Sexual Intimate Partner Violence who Reported

Reasons for Asking for Help With, and for Leaving Violent Relationships

Reasons That Made You Go for Help (n = 486) Percentage (95% CI)

Reasons for Leaving the Last Time (n = 508) Percentage (95% CI)

Could not endure more 48.5 (43.6–53.3) 64.2 (59.8–68.6)

Encouraged by friends/family 17.7 (14.0–21.5) 6.7 (4.4–9.0)

Badly injured 15.4 (12.0–18.7) 7.1 (4.2–9.9)

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Table 7.3 Percentage of Women With Lifetime Experience of Physical and/or Sexual Intimate Partner Violence who Reported

Reasons for Asking for Help With, and for Leaving Violent Relationships

Reasons That Made You Go for Help (n = 486) Percentage (95% CI)

Reasons for Leaving the Last Time (n = 508) Percentage (95% CI)

He threated to kill her 11.3 (8.1–14.5) 10.2 (7.0–13.3)

Afraid he would kill her 11.4 (8.3–14.6) 5.9 (3.8–7.9)

Children suffering 17.2 (13.5–20.9) 8.6 (5.9–11.3)

He threatened or hit children 7.9 (5.2–10.5) 5.0 (3.0–7.0)

She was afraid she would kill him

2.1 (0.8–3.5) 1.2 (0.2–2.2)

She was thrown out of the home

2.0 (0.6–3.3) 2.2 (0.7–3.7)

For her mental health/save sanity

8.2 (5.5–10.9) —

To get information/legal help 8.1 (5.5–10.6) —

Encouraged by organization — 1.7 (0.1–3.3)

Table 7.3 Percentage of Women With Lifetime Experience of Physical and/or Sexual Intimate Partner Violence who Reported

Reasons for Asking for Help With, and for Leaving Violent Relationships

Reasons That Made You Go for Help (n = 486) Percentage (95% CI)

Reasons for Leaving the Last Time (n = 508) Percentage (95% CI)

No particular incident — 5.3 (3.3–7.4)

He was unfaithful — 3.8 (2.1–5.5)

She was pregnant — 1.4 (0.4–2.3)

To have time out/break from relationships

— 1.4 (0.3–2.4)

Other 22.4 (18.5–26.2) 8.3 (5.6–10.9)

Source: Adapted from Janet Fanslow and Elizabeth Robinson, “Help Seeking Behaviors and Reasons for Help Seeking Reported by a Representative Sample of Women Victims of Intimate Partner Violence in New Zealand,” Journal of Interpersonal Violence, 25, no. 5 (2010), 929–951.

Note: Percentages sum to greater than 100% because individuals could provide multiple responses.

Data at Work

Laurel Person Mecca: Research Specialist

Photo courtesy of Laurel Person Mecca

Laurel works as a research specialist in a university center for social and urban research. The center provides support and consultation to behavioral and social science investigators. She first connected with the center while she was recruiting subjects for her master’s thesis research.

“Interacting with human subjects offers a unique view into peoples’ lives,” says Laurel, “thereby providing insights into one’s own life and a richer understanding of the human condition.” She’s consulted on an array of projects: testing prototypes of technologies designed to enable older adults and persons with disabilities to live independently, facilitating focus groups with low-income individuals to explore their barriers to Supplemental Nutrition Assistance Program participation, and evaluating an intervention designed to improve parent-adolescent communication about sexual behaviors to reduce STDs and unintended pregnancies among teens.

For students interested in a career in research and data analysis, she offers this advice: “Gain on-the-job experience while in college, even if it is an unpaid internship. Find researchers who are conducting studies that interest you, and inquire about working for them. Even if they are not posting an available position, they may bring you on board (as I have done with students). Persistence pays off! You are much more likely to be selected for a position if you demonstrate a genuine interest in the work and if you continue to show your enthusiasm by following up. Definitely check out the National Science Foundation’s Research Experience for Undergraduates program. Though most of these internships are in the “hard” sciences, there are plenty of openings in social sciences disciplines. These internships include a stipend (YES!), and oftentimes, assistance with travel and housing. They are wonderful opportunities to work directly on a research project, and may provide the additional benefit of a conference presentation and/or publication.”

Main Points

• The goal of most research is to find population parameters. The major objective of sampling theory and statistical inference is to provide estimates of unknown parameters from sample statistics.

• Researchers make point estimates and interval estimates. Point estimates are sample statistics used to estimate the exact value of a population parameter. Interval estimates are ranges of values within which the population parameter may fall.

• Confidence intervals can be used to estimate population parameters such as means or proportions. Their accuracy is defined with the confidence level. The most common confidence levels are 90%, 95%, and 99%.

• To establish a confidence interval for a mean or a proportion, add or subtract from the mean or the proportion the product of the standard error and the Z value corresponding to the confidence level.

Key Terms

• confidence interval (CI) 180

• confidence level 180

• estimation 180

• margin of error 181

• point estimate 180

Digital Resources

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SPSS Demonstration [GSS14SSDS-B]

Producing Confidence Intervals Around a Mean

SPSS calculates confidence intervals around a sample mean or proportion with the Explore procedure. Activate the Explore procedure by selecting the Analyze menu, Descriptive Statistics, then Explore. The opening dialog box has spaces for both dependent and independent variables.

Place HRSRELAX (after an average work day, about how many hours do you have to relax or pursue the activities you enjoy?) in the Dependent List box, and SEX (respondent’s sex) in the Factor List box, as shown in Figure 7.7. We expect a gender difference in average relaxation hours, with women having less hours than men. The GSS asked respondents about their relaxation time after an average workday. In fact, social scientists have documented how women (not men)

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assume a second shift after work, managing family responsibilities at home, and consequently having less time for themselves.

Figure 7.7 Explore Dialog Box for HRSRELAX and SEX

Click on the Statistics button. Note that the Descriptives choice also includes the confidence interval for the mean, which by default is calculated at the 95% confidence level. Let’s change that to the 99% level by erasing the “95” and substituting “99”.

Click on Continue to return to the main dialog box. Recall that Explore produces several statistics and plots by default. For this example, we don’t need to view the graphics, so click on the Statistics button in the Display section. Click on OK to run the procedure.

The output from the Explore procedure (Figure 7.8) is divided into two parts, one for males and the other for females. The mean number of HRSRELAX for males is 3.68; for females, it’s 3.18. Our data indicate that, on average, men have slightly more time to relax in a day.

The 99% confidence interval for males runs from 3.26 to 4.11 hours. One way to interpret this result is to state that, in 100 samples each of size 244 males from the U.S. adult population, we would expect the confidence interval to include the true population value for the mean ideal number of children 99 times out of those 100. We can never be sure that in this particular sample the confidence interval includes the population mean. Still, our best estimate for the average relaxation hours per day falls within a narrow range of 0.85 (4.11–3.26). For females, the 99% confidence interval is 2.79 to 3.57 hours, with a range of 0.78 (3.57–2.79).

Figure 7.8 SPSS Output for HRSRELAX by SEX

We’ll continue to explore the relationship between gender and relaxation hours in Chapter 8 (“Testing Hypotheses”).

SPSS Problems [GSS14SSDS-b]

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1. Recall that the GSS sample includes men and women from 18 to 89 years of age. Does it matter that we have responses from men and women of diverse ages? Would our HRSRELAX results change if we selected a younger sample of men and women?

a. To take the SPSS demonstration one step further, use the Select Cases procedure to select respondents based on the variable AGE who are less than or equal to 35 years old. Do this by selectingData and then Select Cases. Next, select If Condition is satisfied and then click on If. Find and highlight the variable AGE in the scroll-down box on the left of your screen. Click the arrow next to the scroll-down box. AGE will now appear in the box on the right. Now, tell SPSS that you want to select respondents who are 35 years of age or less. The box on the right should now read AGE <= 35. Click Continue and then OK.

b. Using this younger sample, repeat the Explore procedure (with the 99% confidence interval) that we just completed in the demonstration. What differences exist between men and women in this younger sample on hours per day the respondent has to relax? How do these results compare with those based on the entire sample?

2. Calculate the 90% confidence interval for the following variables, comparing lower, working, middle, and upper classes (CLASS) in the GSS sample. Use the Explore procedure using CLASS as your factor variable (Analyze, Descriptive Statistics, Explore). Make a summary statement of your findings. (Reset Select Cases to All cases before completing this exercise.)

a. HRSRELAX (hours to relax per day) b. EDUC (Respondent’s highest year of school completed) c. HRS1 (work hours per week) d. MAEDUC (Mother’s highest year of school completed)

Chapter Exercises

1. In the 2013 National Crime Victimization Study, the Federal Bureau of Investigation found that 23% of Americans age 12 or older had been victims of crime during a 1-year period. This result was based on a sample of 160,040 persons.12

a. Estimate the percentage of U.S. adults who were victims at the 90% confidence level. Provide an interpretation of the confidence interval.

b. Estimate the percentage of victims at the 99% confidence level. Provide an interpretation of the confidence interval.

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2. Use the GSS 2014 data on education from Chapter 5 (“The Normal Distribution”), Exercise 5.

Mean Standard Deviation N

Lower class 12.11 2.83 122

Working class 13.01 2.91 541

Middle class 14.99 2.93 475

Upper class 15.44 2.83 34

a. Construct the 95% confidence interval for the mean number of years of education for lower-class and working-class respondents.

b. Construct the 99% confidence interval for the mean number of years of education for lower-class and middle-class respondents.

c. As our confidence in the result increases, how does the size of the confidence interval change? Explain why this is so.

3. In 2016, the Pew Research Center conducted a survey of 1,004 Canadians and 1,003 Americans to assess their opinion of climate change. The data show that 51% of Canadians and 45% of Americans believe climate change is a very serious problem.13

a. Estimate the proportion of all Canadians who believe climate change is a very serious problem at the 95% confidence interval.

b. Estimate the proportion of all Americans who believe climate change is a very serious problem at the 95% confidence interval.

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c. Would you be confident in reporting that the majority (51% or higher) of Canadians and Americans believe climate change is a very serious problem? Explain.

4. Though 70% of women with children younger than 18 years participate in the labor force,14 society still upholds the stay-at-home mother as the traditional model. Some believe that employment distracts mothers from their parenting role, affecting the well-being of children.

a. In the GSS 2014, respondents were asked to indicate their level of agreement to the statement, “A working mother hurts children”. Of the 435 male respondents who answered the question, 18% strongly agreed that a working mother does not hurt

children. Construct a 90% confidence interval for this statistic.

b. Of the 566 female respondents who answered the question, 40% strongly agreed that a working mother does not hurt children. Construct a 90% confidence interval for this statistic.

c. Why do you think there is a difference between men and women on this issue? 5. Gallup conducted a survey in April 1 to 25, 2010, to determine the congressional vote preference of the American

voters.15 They found that 51% of the male voters preferred a Republican candidate to a Democratic candidate in a sample of 5,490

registered voters. Gallup asks you, their statistical consultant, to tell them whether you could declare the Republican candidate as the

likely winner of the votes coming from men if there was an election today. What is your advice? Why?

6. You have been doing research for your statistics class on the prevalence of binge drinking among teens.

a. According to 2011 Monitoring the Future data, the average binge drinking score, for this sample of 914 teens, is 1.27, with a standard deviation of 0.80. Binge drinking is defined as the number of times the teen drank five or more alcoholic drinks during

the past week. Construct the 95% confidence interval for the true average severe binge drinking score.

b. Your roommate is concerned about your confidence interval calculation, arguing that severe binge drinking scores are not normally distributed, which in turn makes the confidence interval calculation meaningless. Assume your roommate is correct

about the distribution of severe binge drinking scores. Does that imply that the calculation of a confidence interval is not appropriate?

Why or why not?

7. From the GSS 2014 subsample, we find that 64% of respondents favor the death penalty for murder (N = 1,403).

a. What is the 95% confidence interval for the percentage of the U.S. population who favor the death penalty for murder? b. Without doing any calculations, estimate the lower and upper bounds of 90% and 99% confidence intervals.

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8. A social service agency plans to conduct a survey to determine the mean income of its clients. The director of the agency prefers that you measure the mean income very accurately, to within ±$500. From a sample taken 2 years ago, you estimate that the standard deviation of income for this population is about $5,000. Your job is to figure out the necessary sample size to reduce sampling error to ±$500.

a. Do you need to have an estimate of the current mean income to answer this question? Why or why not? b. What sample size should be drawn to meet the director’s requirement at the 95% level of confidence? (Hint: Use the

formula for a confidence interval and solve for N, the sample size.)

c. What sample size should be drawn to meet the director’s requirement at the 99% level of confidence?

9. Education data (measured in years) from the ISSP 2014 are presented below for four countries. Calculate the 90% confidence interval for each country.

Country Mean Standard Deviation N

France 14.12 5.73 975

Japan 12.48 2.53 528

Croatia 12.18 2.71 480

Turkey 9.15 11.98 783

10. Throughout the 2016 Presidential election primaries, Millennials (those aged 20 to 36 years) consistently supported Senator Bernie Sanders over Secretary Hillary Clinton. According to the 2016 Gallup poll of 1,754 Millennials, 55% had a favorable opinion

of Sanders than Hillary Clinton (38%).16 Calculate the 90% confidence interval for both reported percentages.

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11. According to a 2014 survey by the Pew Research Center, 18% of registered Republicans and 15% of registered Democrats follow political candidates on social media.17 These data are based on a national survey of 446 registered Republicans and 522

registered Democrats. What is the 95% confidence interval for the percentage of Republicans who follow political candidates on

social media? The 95% confidence interval for Democrats?

12. According to a report published by the Pew Research Center in February 2010, 61% of Millennials (Americans in their teens and 20s) think that their generation has a unique and distinctive identity (N = 527).18

a. Calculate the 95% confidence interval to estimate the percentage of Millennials who believe that their generation has a distinctive identity as compared with the other generations (Generation X, baby boomers, or the Silent Generation).

b. Calculate the 99% confidence interval. c. Are both these results compatible with the conclusion that the majority of Millennials believe that they have a unique

identity that separates them from the previous generations?

13. In 2014, GSS respondents (N = 950) were asked whether homosexual relations were wrong. The data show that 40% believed that homosexual relations were always wrong, while 49% believed that homosexual relations were not wrong at all.

a. For each reported percentage, calculate the 95% confidence interval. b. Approximately 10% of GSS respondents were in the middle, some saying that homosexual relations were almost

always wrong or sometimes wrong. Calculate the 95% confidence interval.

c. Based on your calculations, what conclusions can you draw about the public’s opinions of homosexual behavior?