Why is it important that statistical research follow ethical guidelines

StatW4

Answer each question for both Part I and II in an essay format with 5-7 sentences( at least one paragraph). Use references from the attached documents.

Part I

The text describes statistical inference as making "accurate decisions about parameters from statistics." (Kozak, 2014, p. 202); in other words, drawing conclusions about populations based on samples. Statistical methods are vital tool in the conduct of research, but only if they are used in a manner that accurately and honestly depicts the information that the research may reveal (or NOT reveal). Ethical considerations also play a major role in proper conduct of statistical research.

Many professional organizations publish ethical guidelines about the use of statistics and research, which have particular relevance for their fields and professions. The ethical guidelines below are from the American Statistical Association (ASA) and are more general to the use of statistics:

American Statistical Association (April 2016). Ethical guidelines for statistical practice. Report prepared by the Committee on Professional Ethics of the American Statistical Association, Alexandria, VA. Retrieved from: http://www.amstat.org/ASA/Your-Career/Ethical-Guidelines-for-Statistical-Practice.aspx

For this discussion, please answer the following two questions. In your chosen field:

· Why is it important that statistical research follow ethical guidelines?

· What are some potential consequences of not following ethical statistical practice in your field of study (i.e., results used improperly, etc.)?

People use data (statistics) to make all kinds of decisions in their lives. Some examples include (where to live, what college to attend, what college major to select, what foods to buy, what kind of house to buy, what kind of car to buy, what consumer goods to buy, what consumer goods to rent, what newspaper to read, how many children to have, who they want as their friends, health and safety issues, do research for a graduate school program, etc. (I think you get the idea).

· In your life or in the life you want to live, what types of data is most important to you- please name 10 types and list them in order from most important to less important.

Part II

Statistics, Research, and Ethical Considerations

The text describes statistical inference as making "accurate decisions about parameters from statistics." (Kozak, 2014, p. 202); in other words, drawing conclusions about populations based on samples. Statistical methods are vital tool in the conduct of research, but only if they are used in a manner that accurately and honestly depicts the information that the research may reveal (or NOT reveal). Ethical considerations also play a major role in proper conduct of statistical research.

Many professional organizations publish ethical guidelines about the use of statistics and research, which have particular relevance for their fields and professions. The ethical guidelines below are from the American Statistical Association (ASA) and are more general to the use of statistics:

American Statistical Association (April 2016). Ethical guidelines for statistical practice. Report prepared by the Committee on Professional Ethics of the American Statistical Association, Alexandria, VA. Retrieved from: http://www.amstat.org/ASA/Your-Career/Ethical-Guidelines-for-Statistical-Practice.aspx

For this discussion, please answer the following two questions. In your chosen field:

· Why is it important that statistical research follow ethical guidelines?

· What are some potential consequences of not following ethical statistical practice in your field of study (i.e., results used improperly, etc.)?

Reference

Kozak, K. (2014). Statistics using technology. Retrieved from https://s3-us-west-2.amazonaws.com/oerfiles/statsusingtech.pdf

Note: If you go above my bid, I will NOT work with you. Don't bother bidding, I DO NOT care about your ratings, I just care that you can get me a good grade. Also I DO NOT RATE bids. That a false representation of how good you are.

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Chapter 6: Continuous Probability Distributions Chapter 5 dealt with probability distributions arising from discrete random variables. Mostly that chapter focused on the binomial experiment. There are many other experiments from discrete random variables that exist but are not covered in this book. Chapter 6 deals with probability distributions that arise from continuous random variables. The focus of this chapter is a distribution known as the normal distribution, though realize that there are many other distributions that exist. A few others are examined in future chapters. Section 6.1: Uniform Distribution If you have a situation where the probability is always the same, then this is known as a uniform distribution. An example would be waiting for a commuter train. The commuter trains on the Blue and Green Lines for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak hours of ten minutes ("2012 annual report," 2012). If you are waiting for a train, you have anywhere from zero minutes to ten minutes to wait. Your probability of having to wait any number of minutes in that interval is the same. This is a uniform distribution. The graph of this distribution is in figure #6.1.1. Figure #6.1.1: Uniform Distribution Graph

Suppose you want to know the probability that you will have to wait between five and ten minutes for the next train. You can look at the probability graphically such as in figure #6.1.2.

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Figure #6.1.2: Uniform Distribution with P(5 < x < 10)

How would you find this probability? Calculus says that the probability is the area under the curve. Notice that the shape of the shaded area is a rectangle, and the area of a rectangle is length times width. The length is 10 − 5 = 5 and the width is 0.1. The probability is P 5 < x <10( ) = 0.1*5 = 0.5 , where and x is the waiting time during peak hours. Example #6.1.1: Finding Probabilities in a Uniform Distribution

The commuter trains on the Blue and Green Lines for the Regional Transit Authority (RTA) in Cleveland, OH, have a waiting time during peak rush hour periods of ten minutes ("2012 annual report," 2012). a.) State the random variable. Solution: x = waiting time during peak hours

b.) Find the probability that you have to wait between four and six minutes for a

train. Solution: P 4 < x < 6( ) = 6 − 4( )*0.1= 0.2 c.) Find the probability that you have to wait between three and seven minutes for

a train. Solution: P 3< x < 7( ) = 7 − 3( )*0.1= 0.4 d.) Find the probability that you have to wait between zero and ten minutes for a

train. Solution: P 0 < x <10( ) = 10 − 0( )*0.1= 1.0

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e.) Find the probability of waiting exactly five minutes. Solution:

Since this would be just one line, and the width of the line is 0, then the P x = 5( ) = 0*0.1= 0

Notice that in example #6.1.1d, the probability is equal to one. This is because the probability that was computed is the area under the entire curve. Just like in discrete probability distributions, where the total probability was one, the probability of the entire curve is one. This is the reason that the height of the curve is 0.1. In general, the height

of a uniform distribution that ranges between a and b, is 1 b − a

.

Section 6.1: Homework 1.) The commuter trains on the Blue and Green Lines for the Regional Transit

Authority (RTA) in Cleveland, OH, have a waiting time during peak rush hour periods of ten minutes ("2012 annual report," 2012). a.) State the random variable. b.) Find the probability of waiting between two and five minutes. c.) Find the probability of waiting between seven and ten minutes. d.) Find the probability of waiting eight minutes exactly.

2.) The commuter trains on the Red Line for the Regional Transit Authority (RTA) in

Cleveland, OH, have a waiting time during peak rush hour periods of eight minutes ("2012 annual report," 2012). a.) State the random variable. b.) Find the height of this uniform distribution. c.) Find the probability of waiting between four and five minutes. d.) Find the probability of waiting between three and eight minutes. e.) Find the probability of waiting five minutes exactly.

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Section 6.2: Graphs of the Normal Distribution Many real life problems produce a histogram that is a symmetric, unimodal, and bell- shaped continuous probability distribution. For example: height, blood pressure, and cholesterol level. However, not every bell shaped curve is a normal curve. In a normal curve, there is a specific relationship between its “height” and its “width.” Normal curves can be tall and skinny or they can be short and fat. They are all symmetric, unimodal, and centered at µ , the population mean. Figure #6.2.1 shows two different normal curves drawn on the same scale. Both have µ = 100 but the one on the left has a standard deviation of 10 and the one on the right has a standard deviation of 5. Notice that the larger standard deviation makes the graph wider (more spread out) and shorter. Figure #6.2.1: Different Normal Distribution Graphs

Every normal curve has common features. These are detailed in figure #6.2.2. Figure #6.2.2: Typical Graph of a Normal Curve

• The center, or the highest point, is at the population mean, µ . • The transition points (inflection points) are the places where the curve changes

from a “hill” to a “valley”. The distance from the mean to the transition point is one standard deviation, σ .

• The area under the whole curve is exactly 1. Therefore, the area under the half below or above the mean is 0.5.

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The equation that creates this curve is f x( ) = 1

σ 2π e −1

2 x−µ σ

⎛ ⎝⎜

⎞ ⎠⎟

2

.

Just as in a discrete probability distribution, the object is to find the probability of an event occurring. However, unlike in a discrete probability distribution where the event can be a single value, in a continuous probability distribution the event must be a range. You are interested in finding the probability of x occurring in the range between a and b, or P a ≤ x ≤ b( ) = P a < x < b( ) . Calculus tells us that to find this you find the area under the curve above the interval from a to b.

P a ≤ x ≤ b( ) = P a < x < b( ) is the area under the curve above the interval from a to b.

Figure #6.2.3: Probability of an Event

Before looking at the process for finding the probabilities under the normal curve, it is somewhat useful to look at the Empirical Rule that gives approximate values for these areas. The Empirical Rule is just an approximation and it will only be used in this section to give you an idea of what the size of the probabilities is for different shadings. A more precise method for finding probabilities for the normal curve will be demonstrated in the next section. Please do not use the empirical rule except for real rough estimates. The Empirical Rule for any normal distribution: Approximately 68% of the data is within one standard deviation of the mean. Approximately 95% of the data is within two standard deviations of the mean.

Approximately 99.7% of the data is within three standard deviations of the mean.

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Figure #6.2.4: Empirical Rule

Be careful, there is still some area left over in each end. Remember, the maximum a probability can be is 100%, so if you calculate 100%− 99.7% = 0.3% you will see that for both ends together there is 0.3% of the curve. Because of symmetry, you can divide this equally between both ends and find that there is 0.15% in each tail beyond the µ ± 3σ .

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Section 6.3: Finding Probabilities for the Normal Distribution The Empirical Rule is just an approximation and only works for certain values. What if you want to find the probability for x values that are not integer multiples of the standard deviation? The probability is the area under the curve. To find areas under the curve, you need calculus. Before technology, you needed to convert every x value to a standardized number, called the z-score or z-value or simply just z. The z-score is a measure of how many standard deviations an x value is from the mean. To convert from a normally distributed x value to a z-score, you use the following formula. z-score

z = x − µ σ

where µ = mean of the population of the x value and σ = standard deviation for the population of the x value The z-score is normally distributed, with a mean of 0 and a standard deviation of 1. It is known as the standard normal curve. Once you have the z-score, you can look up the z- score in the standard normal distribution table. The standard normal distribution, z, has a mean of µ = 0 and a standard deviation of σ = 1. Figure #6.3.1: Standard Normal Curve

Luckily, these days technology can find probabilities for you without converting to the z- score and looking the probabilities up in a table. There are many programs available that will calculate the probability for a normal curve including Excel and the TI-83/84. There are also online sites available. The following examples show how to do the calculation on the TI-83/84. The command on the TI-83/84 is in the DISTR menu and is normalcdf(. You then type in the lower limit, upper limit, mean, standard deviation in that order and including the commas.

z

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Example #6.3.1: General Normal Distribution The length of a human pregnancy is normally distributed with a mean of 272 days with a standard deviation of 9 days (Bhat & Kushtagi, 2006). a.) State the random variable. Solution:

x = length of a human pregnancy b.) Find the probability of a pregnancy lasting more than 280 days. Solution:

First translate the statement into a mathematical statement. P x > 280( )

Now, draw a picture. Remember the center of this normal curve is 272.

Figure #6.3.2: Normal Distribution Graph for Example #6.3.1b

Looking at the picture, you realize the lower limit is 280. The upper limit is infinity. The calculator doesn’t have infinity on it, so you need to put in a really big number. Some people like to put in 1000, but if you are working with numbers that are bigger than 1000, then you would have to remember to change the upper limit. The safest number to use is 1×1099 , which you put in the calculator as 1E99 (where E is the EE button on the calculator). The command looks like:

normalcdf 280,1E99,272,9( ) Figure #6.3.3: TI-83/84 Output for Example #6.3.1b

Thus, P x > 280( ) ≈ 0.187

Thus18.7% of all pregnancies last more than 280 days. This is not unusual since the probability is greater than 5%.

0

0.01

0.02

0.03

0.04

0.05

220 240 260 280 300 320

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c.) Find the probability of a pregnancy lasting less than 250 days. Solution:

First translate the statement into a mathematical statement. P x < 250( )

Now, draw a picture. Remember the center of this normal curve is 272.

Figure #6.3.4: Normal Distribution Graph for Example #6.3.1c

Looking at the picture, though it is hard to see in this case, the lower limit is negative infinity. Again, the calculator doesn’t have this on it, put in a really small number, such as −1×1099 = −1E99 on the calculator.

Figure #6.3.5: TI-83/84 Output for Example #6.3.1c

P x < 250( ) = normalcdf −1E99,250,272,9( ) = 0.0073 .

Thus 0.73% of all pregnancies last less than 250 days. This is unusual since the probability is less than 5%.

d.) Find the probability that a pregnancy lasts between 265 and 280 days. Solution:

First translate the statement into a mathematical statement. P 265 < x < 280( )

Now, draw a picture. Remember the center of this normal curve is 272.

0

0.01

0.02

0.03

0.04

0.05

230 240 250 260 270 280 290 300 310 320

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Figure #6.3.6: Normal Distribution Graph for Example #6.3.1d

In this case, the lower limit is 265 and the upper limit is 280. Using the calculator

Figure #6.3.7: TI-83/84 Output for Example #6.3.1d

P 265 < x < 280( ) = normalcdf 265,280,272,9( ) = 0.595

Thus 59.5% of all pregnancies last between 265 and 280 days.

e.) Find the length of pregnancy that 10% of all pregnancies last less than. Solution:

This problem is asking you to find an x value from a probability. You want to find the x value that has 10% of the length of pregnancies to the left of it. There is a command on the TI-83/84 in the DISTR menu. It is called invNorm(. The invNorm( command needs the area to the left. In this case, that is the area you are given. For the command on the calculator, once you have invNorm( on the main screen you type in the probability to the left, mean, standard deviation, in that order with the commas. Figure #6.3.8: TI-83/84 Output for Example #6.3.1e

Thus 10% of all pregnancies last less than approximately 260 days.

0

0.01

0.02

0.03

0.04

0.05

230 240 250 260 270 280 290 300 310 320

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f.) Suppose you meet a woman who says that she was pregnant for less than 250 days. Would this be unusual and what might you think?

Solution:

From part (c) you found the probability that a pregnancy lasts less than 250 days is 0.73%. Since this is less than 5%, it is very unusual. You would think that either the woman had a premature baby, or that she may be wrong about when she actually became pregnant.

Example #6.3.2: General Normal Distribution

The mean mathematics SAT score in 2012 was 514 with a standard deviation of 117 ("Total group profile," 2012). Assume the mathematics SAT score is normally distributed. a.) State the random variable. Solution: x = mathematics SAT score

b.) Find the probability that a person has a mathematics SAT score over 700. Solution:

First translate the statement into a mathematical statement. P x > 700( )

Now, draw a picture. Remember the center of this normal curve is 514.

Figure #6.3.9: Normal Distribution Graph for Example #6.3.2b

P x > 700( ) = normalcdf 700,1E99,514,117( ) ≈ 0.056

There is a 5.6% chance that a person scored above a 700 on the mathematics SAT test. This is not unusual.

c.) Find the probability that a person has a mathematics SAT score of less than 400. Solution:

First translate the statement into a mathematical statement. P x < 400( )

Now, draw a picture. Remember the center of this normal curve is 514.

0 0.0005 0.001

0.0015 0.002

0.0025 0.003

0.0035 0.004

100 200 300 400 500 600 700 800 900

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Figure #6.3.10: Normal Distribution Graph for Example #6.3.2c

P x < 400( ) = normalcdf −1E99,400,514,117( ) ≈ 0.165

So, there is a 16.5% chance that a person scores less than a 400 on the mathematics part of the SAT.

d.) Find the probability that a person has a mathematics SAT score between a 500

and a 650. Solution:

First translate the statement into a mathematical statement. P 500 < x < 650( )

Now, draw a picture. Remember the center of this normal curve is 514.

Figure #6.3.11: Normal Distribution Graph for Example #6.3.2d

P 500 < x < 650( ) = normalcdf 500,650,514,117( ) ≈ 0.425

So, there is a 42.5% chance that a person has a mathematical SAT score between 500 and 650.

e.) Find the mathematics SAT score that represents the top 1% of all scores.

Solution:

This problem is asking you to find an x value from a probability. You want to find the x value that has 1% of the mathematics SAT scores to the right of it. Remember, the calculator always needs the area to the left, you need to find the area to the left by 1− 0.01= 0.99 . Now use invNorm(

invNorm .99,514,117( ) ≈ 786 So, 1% of all people who took the SAT scored over about 786 points on the mathematics SAT.

0 0.0005 0.001

0.0015 0.002

0.0025 0.003

0.0035 0.004

0 200 400 600 800 1000 1200

0 0.0005 0.001

0.0015 0.002

0.0025 0.003

0.0035 0.004

0 200 400 600 800 1000 1200

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Section 6.3: Homework 1.) Find each of the probabilities, where z is a z-score from the standard normal

distribution with mean of µ = 0 and standard deviation σ = 1. Make sure you draw a picture for each problem. a.) P z < 2.36( ) b.) P z > 0.67( ) c.) P 0 < z < 2.11( ) d.) P −2.78 < z <1.97( )

2.) Find the z-score corresponding to the given area. Remember, z is distributed as

the standard normal distribution with mean of µ = 0 and standard deviation σ = 1. a.) The area to the left of z is 15%. b.) The area to the right of z is 65%. c.) The area to the left of z is 10%. d.) The area to the right of z is 5%. e.) The area between −z and z is 95%. (Hint draw a picture and figure out the

area to the left of the −z .) f.) The area between −z and z is 99%.

3.) If a random variable that is normally distributed has a mean of 25 and a standard

deviation of 3, convert the given value to a z-score. a.) x = 23 b.) x = 33 c.) x = 19 d.) x = 45

4.) According to the WHO MONICA Project the mean blood pressure for people in

China is 128 mmHg with a standard deviation of 23 mmHg (Kuulasmaa, Hense & Tolonen, 1998). Assume that blood pressure is normally distributed. a.) State the random variable. b.) Find the probability that a person in China has blood pressure of 135 mmHg

or more. c.) Find the probability that a person in China has blood pressure of 141 mmHg

or less. d.) Find the probability that a person in China has blood pressure between 120

and 125 mmHg. e.) Is it unusual for a person in China to have a blood pressure of 135 mmHg?

Why or why not? f.) What blood pressure do 90% of all people in China have less than?

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5.) The size of fish is very important to commercial fishing. A study conducted in 2012 found the length of Atlantic cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm (Ovegard, Berndt & Lunneryd, 2012). Assume the length of fish is normally distributed. a.) State the random variable. b.) Find the probability that an Atlantic cod has a length less than 52 cm. c.) Find the probability that an Atlantic cod has a length of more than 74 cm. d.) Find the probability that an Atlantic cod has a length between 40.5 and 57.5

cm. e.) If you found an Atlantic cod to have a length of more than 74 cm, what could

you conclude? f.) What length are 15% of all Atlantic cod longer than?

6.) The mean cholesterol levels of women age 45-59 in Ghana, Nigeria, and Seychelles is 5.1 mmol/l and the standard deviation is 1.0 mmol/l (Lawes, Hoorn, Law & Rodgers, 2004). Assume that cholesterol levels are normally distributed. a.) State the random variable. b.) Find the probability that a woman age 45-59 in Ghana, Nigeria, or Seychelles

has a cholesterol level above 6.2 mmol/l (considered a high level). c.) Find the probability that a woman age 45-59 in Ghana, Nigeria, or Seychelles

has a cholesterol level below 5.2 mmol/l (considered a normal level). d.) Find the probability that a woman age 45-59 in Ghana, Nigeria, or Seychelles

has a cholesterol level between 5.2 and 6.2 mmol/l (considered borderline high).

e.) If you found a woman age 45-59 in Ghana, Nigeria, or Seychelles having a cholesterol level above 6.2 mmol/l, what could you conclude?

f.) What value do 5% of all woman ages 45-59 in Ghana, Nigeria, or Seychelles have a cholesterol level less than?

7.) In the United States, males between the ages of 40 and 49 eat on average 103.1 g

of fat every day with a standard deviation of 4.32 g ("What we eat," 2012). Assume that the amount of fat a person eats is normally distributed. a.) State the random variable. b.) Find the probability that a man age 40-49 in the U.S. eats more than 110 g of

fat every day. c.) Find the probability that a man age 40-49 in the U.S. eats less than 93 g of fat

every day. d.) Find the probability that a man age 40-49 in the U.S. eats less than 65 g of fat

every day. e.) If you found a man age 40-49 in the U.S. who says he eats less than 65 g of fat

every day, would you believe him? Why or why not? f.) What daily fat level do 5% of all men age 40-49 in the U.S. eat more than?

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8.) A dishwasher has a mean life of 12 years with an estimated standard deviation of 1.25 years ("Appliance life expectancy," 2013). Assume the life of a dishwasher is normally distributed. a.) State the random variable. b.) Find the probability that a dishwasher will last more than 15 years. c.) Find the probability that a dishwasher will last less than 6 years. d.) Find the probability that a dishwasher will last between 8 and 10 years. e.) If you found a dishwasher that lasted less than 6 years, would you think that

you have a problem with the manufacturing process? Why or why not? f.) A manufacturer of dishwashers only wants to replace free of charge 5% of all

dishwashers. How long should the manufacturer make the warranty period?

9.) The mean starting salary for nurses is $67,694 nationally ("Staff nurse -," 2013). The standard deviation is approximately $10,333. Assume that the starting salary is normally distributed. a.) State the random variable. b.) Find the probability that a starting nurse will make more than $80,000. c.) Find the probability that a starting nurse will make less than $60,000. d.) Find the probability that a starting nurse will make between $55,000 and

$72,000. e.) If a nurse made less than $50,000, would you think the nurse was under paid?

Why or why not? f.) What salary do 30% of all nurses make more than?

10.) The mean yearly rainfall in Sydney, Australia, is about 137 mm and the standard deviation is about 69 mm ("Annual maximums of," 2013). Assume rainfall is normally distributed. a.) State the random variable. b.) Find the probability that the yearly rainfall is less than 100 mm. c.) Find the probability that the yearly rainfall is more than 240 mm. d.) Find the probability that the yearly rainfall is between 140 and 250 mm. e.) If a year has a rainfall less than 100mm, does that mean it is an unusually dry

year? Why or why not? f.) What rainfall amount are 90% of all yearly rainfalls more than?

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Section 6.4: Assessing Normality The distributions you have seen up to this point have been assumed to be normally distributed, but how do you determine if it is normally distributed. One way is to take a sample and look at the sample to determine if it appears normal. If the sample looks normal, then most likely the population is also. Here are some guidelines that are use to help make that determination.

1. Histogram: Make a histogram. For a normal distribution, the histogram should be roughly bell-shaped. For small samples, this is not very accurate, and another method is needed. A distribution may not look normally distributed from the histogram, but it still may be normally distributed.

2. Outliers: For a normal distribution, there should not be more than one outlier. One way to check for outliers is to use a box plot. Outliers are those data values that are:

Above Q3, the third quartile, by an amount greater than 1.5 times the interquartile range (IQR) Below Q1, the first quartile, by an amount greater than 1.5 times the interquartile range (IQR)

Note: if there is one outlier, that outlier could have a dramatic effect on the results especially if it is an extreme outlier. However, there are times where a distribution has more than one outlier, but it is still normally distributed. The guideline of only one outlier is just a guideline.

3. Normal probability plot (or normal quantile plot): This plot is provided through statistical software on a computer or graphing calculator. If the points lie close to a straight line, the data comes from a distribution that is approximately normal. If the points do not lie close to a straight line or they show a pattern that is not a straight line, the data are likely to come from a distribution that is not normally distributed.

To create a histogram on the TI-83/84:

1. Go into the STAT menu, and then Chose 1:Edit Figure #6.4.1: STAT Menu on TI-83/84

2. Type your data values into L1.

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3. Now click STAT PLOT (2nd Y=).

Figure #6.4.2: STAT PLOT Menu on TI-83/84

4. Use 1:Plot1. Press ENTER. Figure #6.4.3: Plot1 Menu on TI-83/84

5. You will see a new window. The first thing you want to do is turn the plot on. At this point you should be on On, just press ENTER. It will make On dark.

6. Now arrow down to Type: and arrow right to the graph that looks like a histogram (3rd one from the left in the top row).

7. Now arrow down to Xlist. Make sure this says L1. If it doesn’t, then put L1 there (2nd number 1). Freq: should be a 1.

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Figure #6.4.4: Plot1 Menu on TI-83/84 Setup for Histogram

8. Now you need to set up the correct window to graph on. Click on WINDOW. You need to set up the settings for the x variable. Xmin should be your smallest data value. Xmax should just be a value sufficiently above your highest data value, but not too high. Xscl is your class width that you calculated. Ymin should be 0 and Ymax should be above what you think the highest frequency is going to be. You can always change this if you need to. Yscl is just how often you would like to see a tick mark on the y-axis.

9. Now press GRAPH. You will see a histogram.

To find the IQR and create a box plot on the TI-83/84: 1. Go into the STAT menu, and then Chose 1:Edit

Figure #6.4.5: STAT Menu on TI-83/84

2. Type your data values into L1. If L1 has data in it, arrow up to the name L1, click CLEAR and then press ENTER. The column will now be cleared and you can type the data in.

3. Go into the STAT menu, move over to CALC and choose 1-Var Stats. Press ENTER, then type L1 (2nd 1) and then ENTER. This will give you the summary statistics. If you press the down arrow, you will see the five-number summary.

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4. To draw the box plot press 2nd STAT PLOT.

Figure #6.4.6: STAT PLOT Menu on TI-83/84

5. Use Plot1. Press ENTER Figure #6.4.7: Plot1 Menu on TI-83/84 Setup for Box Plot

6. Put the cursor on On and press Enter to turn the plot on. Use the down arrow and the right arrow to highlight the boxplot in the middle of the second row of types then press ENTER. Set Data List to L1 (it might already say that) and leave Freq as 1.

7. Now tell the calculator the set up for the units on the x-axis so you can see the whole plot. The calculator will do it automatically if you press ZOOM, which is in the middle of the top row.

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Figure #6.4.8: ZOOM Menu on TI-83/84

Then use the down arrow to get to 9:ZoomStat and press ENTER. The box plot will be drawn. Figure #6.4.9: ZOOM Menu on TI-83/84 with ZoomStat

To create a normal probability plot on the TI-83/84: 1. Go into the STAT menu, and then Chose 1:Edit

Figure #6.4.10: STAT Menu on TI-83/84

2. Type your data values into L1. If L1 has data in it, arrow up to the name L1, click CLEAR and then press ENTER. The column will now be cleared and you can type the data in.

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3. Now click STAT PLOT (2nd Y=). You have three stat plots to choose from.

Figure #6.4.11: STAT PLOT Menu on TI-83/84

4. Use 1:Plot1. Press ENTER.

5. Put the curser on the word On and press ENTER. This turns on the plot. Arrow down to Type: and use the right arrow to move over to the last graph (it looks like an increasing linear graph). Set Data List to L1 (it might already say that) and set Data Axis to Y. The Mark is up to you. Figure #6.4.12: Plot1 Menu on TI-83/84 Setup for Normal Probability Plot

6. Now you need to set up the correct window on which to graph. Click on WINDOW. You need to set up the settings for the x variable. Xmin should be −4 . Xmax should be 4. Xscl should be 1. Ymin and Ymax are based on your data, the Ymin should be below your lowest data value and Ymax should be above your highest data value. Yscl is just how often you would like to see a tick mark on the y-axis.

7. Now press GRAPH. You will see the normal probability plot. Realize that your random variable may be normally distributed, even if the sample fails the three tests. However, if the histogram definitely doesn't look symmetric and bell shaped, there are outliers that are very extreme, and the normal probability plot doesn’t

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look linear, then you can be fairly confident that the data set does not come from a population that is normally distributed. Example #6.4.1: Is It Normal?

In Kiama, NSW, Australia, there is a blowhole. The data in table #6.4.1 are times in seconds between eruptions ("Kiama blowhole eruptions," 2013). Do the data come from a population that is normally distributed?

Table #6.4.1: Time (in Seconds) Between Kiama Blowhole Eruptions

83 51 87 60 28 95 8 27 15 10 18 16 29 54 91 8 17 55 10 35 47 77 36 17 21 36 18 40 10 7 34 27 28 56 8 25 68 146 89 18 73 69 9 37 10 82 29 8 60 61 61 18 169 25 8 26 11 83 11 42 17 14 9 12

a.) State the random variable

Solution: x = time in seconds between eruptions of Kiama Blowhole

b.) Draw a histogram. The class width is 169 − 7 7

≈ 23.1≈ 24

Solution:

Drawing the histogram on the TI-83/84 produces figure #6.4.13. Figure #6.4.13: Histogram for Kiama Blowhole

This looks skewed right and not symmetric.

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c.) Find the number of outliers. Solution:

The five-number summary from the TI-83/84 is in figure #6.4.14. Figure #6.4.14: Output from TI-83/84 for Kiama Blowhole

IQR =Q3−Q1= 60 −14.5 = 45.5 seconds 1.5 * IQR = 1.5 * 45.5 = 68.25 seconds Q1−1.5 * IQR = 14.5 − 68.25 = −53.75 seconds Q3+1.5 * IQR = 60 + 68.25 = 128.25 seconds

Outliers are any numbers greater than 128.25 seconds and less than −53.75 seconds. Since all the numbers are measurements of time, then no data values are less than 0 or −53.75 seconds for that matter. There are two numbers that are larger than 128.25 seconds, so there are two outliers. Two outliers are not real indications that the sample does not come from a normal distribution, but the fact that both are well above 128.25 seconds is an indication of an issue.

d.) Draw the normal probability plot.

Solution: Drawing the normal probability plot on the TI-83/84 produces figure #6.4.15.

Figure #6.4.15: Normal Probability Plot

This graph looks more like an exponential growth than linear.

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e.) Do the data come from a population that is normally distributed?

Solution: Considering the histogram is skewed right, there are two extreme outliers, and the normal probability plot does not look linear, then the conclusion is that this sample is not from a population that is normally distributed.

Example #6.4.2: Is It Normal?

One way to measure intelligence is with an IQ score. Table #6.4.2 contains 50 IQ scores. Determine if the sample comes from a population that is normally distributed.

Table #6.4.2: IQ Scores

78 92 96 100 67 105 109 75 127 111 93 114 82 100 125 67 94 74 81 98

102 108 81 96 103 91 90 96 86 92 84 92 90 103 115 93 85 116 87 106 85 88 106 104 102 98 116 107 102 89

a.) State the random variable

Solution: x = IQ score

b.) Draw a histogram. The class width is 127 − 67 7

≈ 8.6 ≈ 9

Solution: The histogram created on the TI-83/84 is in figure #6.4.16. Figure #6.4.16: Histogram for IQ Score

This looks somewhat symmetric, though it could be thought of as slightly skewed right.

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c.) Find the number of outliers. Solution:

The five-number summary from the TI-83/84 is in figure #6.4.17. Figure #6.4.17: Output from TI-83/84 for IQ Score

IQR =Q3−Q1= 105 − 87 = 18 1.5* IQR = 1.5*18 = 27 Q1−1.5IQR = 87 − 27 = 60 Q3+1.5IQR = 105 + 27 = 132

Outliers are any numbers greater than 132 and less than 60. Since the maximum number is 127 and the minimum is 67, there are no outliers.

d.) Draw the normal probability plot.

Solution: Drawing the normal probability plot on the TI-83/84 produces figure #6.4.18.

Figure #6.4.18: Normal Probability Plot

This graph looks fairly linear.

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e.) Do the data come from a population that is normally distributed? Solution:

Considering the histogram is somewhat symmetric, there are no outliers, and the normal probability plot looks linear, then the conclusion is that this sample is from a population that is normally distributed.

Section 6.4: Homework 1.) Cholesterol data was collected on patients four days after having a heart attack.

The data is in table #6.4.3. Determine if the data is from a population that is normally distributed. Table #6.4.3: Cholesterol Data Collected Four Days After a Heart Attack

218 234 214 116 200 276 146 182 238 288 190 236 244 258 240 294 220 200 220 186 352 202 218 248 278 248 270 242

2.) The size of fish is very important to commercial fishing. A study conducted in

2012 collected the lengths of Atlantic cod caught in nets in Karlskrona (Ovegard, Berndt & Lunneryd, 2012). Data based on information from the study is in table #6.4.4. Determine if the data is from a population that is normally distributed. Table #6.4.4: Atlantic Cod Lengths

48 50 50 55 53 50 49 52 61 48 45 47 53 46 50 48 42 44 50 60 54 48 50 49 53 48 52 56 46 46 47 48 48 49 52 47 51 48 45 47

3.) The WHO MONICA Project collected blood pressure data for people in China (Kuulasmaa, Hense & Tolonen, 1998). Data based on information from the study is in table #6.4.5. Determine if the data is from a population that is normally distributed. Table #6.4.5: Blood Pressure Values for People in China

114 141 154 137 131 132 133 156 119 138 86 122 112 114 177 128 137 140 171 129 127 104 97 135 107 136 118 92 182 150 142 97 140 106 76 115

119 125 162 80 138 124 132 143 119

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4.) Annual rainfalls for Sydney, Australia are given in table #6.4.6. ("Annual maximums of," 2013). Can you assume rainfall is normally distributed? Table #6.4.6: Annual Rainfall in Sydney, Australia

146.8 383 90.9 178.1 267.5 95.5 156.5 180 90.9 139.7 200.2 171.7 187.2 184.9 70.1 58 84.1 55.6 133.1 271.8 135.9 71.9 99.4 110.6 47.5 97.8 122.7 58.4 154.4 173.7 118.8 88 84.6 171.5 254.3 185.9 137.2 138.9 96.2 85 45.2 74.7 264.9 113.8 133.4 68.1 156.4

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Section 6.5: Sampling Distribution and the Central Limit Theorem You now have most of the skills to start statistical inference, but you need one more concept. First, it would be helpful to state what statistical inference is in more accurate terms. Statistical Inference: to make accurate decisions about parameters from statistics When it says “accurate decision,” you want to be able to measure how accurate. You measure how accurate using probability. In both binomial and normal distributions, you needed to know that the random variable followed either distribution. You need to know how the statistic is distributed and then you can find probabilities. In other words, you need to know the shape of the sample mean or whatever statistic you want to make a decision about. How is the statistic distributed? This is answered with a sampling distribution. Sampling Distribution: how a sample statistic is distributed when repeated trials of size n are taken. Example #6.5.1: Sampling Distribution

Suppose you throw a penny and count how often a head comes up. The random variable is x = number of heads. The probability distribution (pdf) of this random variable is presented in figure #6.5.1.

Figure #6.5.1: Distribution of Random Variable

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Repeat this experiment 10 times, which means n = 10. Here is the data set: {1, 1, 1, 1, 0, 0, 0, 0, 0, 0}. The mean of this sample is 0.4. Now take another sample. Here is that data set: {1, 1, 1, 0, 1, 0, 1, 1, 0, 0}. The mean of this sample is 0.6. Another sample looks like: {0, 1, 0, 1, 1, 1, 1, 1, 0, 1}. The mean of this sample is 0.7. Repeat this 40 times. You could get these means:

Table #6.5.1: Sample Means When n = 10

0.4 0.6 0.7 0.3 0.3 0.2 0.5 0.5 0.5 0.5 0.4 0.4 0.5 0.7 0.7 0.6 0.4 0.4 0.4 0.6 0.7 0.7 0.3 0.5 0.6 0.3 0.3 0.8 0.3 0.6 0.4 0.3 0.5 0.6 0.5 0.6 0.3 0.5 0.6 0.2

Table #6.5.2 contains the distribution of these sample means (just count how many of each number there are and then divide by 40 to obtain the relative frequency).

Table #6.5.2: Distribution of Sample Means When n = 10

Sample Mean Probability 0.1 0 0.2 0.05 0.3 0.2 0.4 0.175 0.5 0.225 0.6 0.2 0.7 0.125 0.8 0.025 0.9 0

Figure #6.5.2 contains the histogram of these sample means.

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Figure #6.5.2: Histogram of Sample Means When n = 10

This distribution (represented graphically by the histogram) is a sampling distribution. That is all a sampling distribution is. It is a distribution created from statistics.

Notice the histogram does not look anything like the histogram of the original random variable. It also doesn’t look anything like a normal distribution, which is the only one you really know how to find probabilities. Granted you have the binomial, but the normal is better.

What does this distribution look like if instead of repeating the experiment 10 times you repeat it 20 times instead?

Table #6.5.3 contains 40 means when the experiment of flipping the coin is repeated 20 times. Table #6.5.3: Sample Means When n = 20 0.5 0.45 0.7 0.55 0.65 0.6 0.4 0.35 0.45 0.6 0.5 0.5 0.65 0.5 0.5 0.35 0.55 0.4 0.65 0.3 0.4 0.5 0.45 0.45 0.65 0.7 0.6 0.5 0.7 0.7 0.7 0.45 0.35 0.6 0.65 0.55 0.35 0.4 0.55 0.6

Table #6.5.3 contains the sampling distribution of the sample means.

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Table #6.5.3: Distribution of Sample Means When n = 20 Mean Probability 0.1 0 0.2 0 0.3 0.125 0.4 0.2 0.5 0.3 0.6 0.25 0.7 0.125 0.8 0 0.9 0

This histogram of the sampling distribution is displayed in figure #6.5.3. Figure #6.5.3: Histogram of Sample Means When n = 20

Notice this histogram of the sample mean looks approximately symmetrical and could almost be called normal. What if you keep increasing n? What will the sampling distribution of the sample mean look like? In other words, what does the sampling distribution of x look like as n gets even larger?

This depends on how the original distribution is distributed. In Example #6.5.1, the random variable was uniform looking. But as n increased to 20, the distribution of the mean looked approximately normal. What if the original distribution was normal? How big would n have to be? Before that question is answered, another concept is needed.

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Suppose you have a random variable that has a population mean, µ , and a population standard deviation, σ . If a sample of size n is taken, then the sample mean, x has a

mean µx = µ and standard deviation of σ x = σ n

. The standard deviation of x is lower

because by taking the mean you are averaging out the extreme values, which makes the distribution of the original random variable spread out. You now know the center and the variability of x . You also want to know the shape of the distribution of x . You hope it is normal, since you know how to find probabilities using the normal curve. The following theorem tells you the requirement to have x normally distributed. Theorem #6.5.1: Central Limit Theorem. Suppose a random variable is from any distribution. If a sample of size n is taken, then the sample mean, x , becomes normally distributed as n increases. What this says is that no matter what x looks like, x would look normal if n is large enough. Now, what size of n is large enough? That depends on how x is distributed in the first place. If the original random variable is normally distributed, then n just needs to be 2 or more data points. If the original random variable is somewhat mound shaped and symmetrical, then n needs to be greater than or equal to 30. Sometimes the sample size can be smaller, but this is a good rule of thumb. The sample size may have to be much larger if the original random variable is really skewed one way or another. Now that you know when the sample mean will look like a normal distribution, then you can find the probability related to the sample mean. Remember that the mean of the sample mean is just the mean of the original data (µx = µ ), but the standard deviation of

the sample mean, σ x , also known as the standard error of the mean, is actually σ x = σ n

.

Make sure you use this in all calculations. If you are using the z-score, the formula when

working with x is z = x − µx σ x

= x − µ σ n

. If you are using the calculator, then the input

would be normalcdf lower limit, upper limit, µ,σ n( ) .

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Example #6.5.1: Finding Probabilities for Sample Means The birth weight of boy babies of European descent who were delivered at 40 weeks is normally distributed with a mean of 3687.6 g with a standard deviation of 410.5 g (Janssen, Thiessen, Klein, Whitfield, MacNab & Cullis-Kuhl, 2007). Suppose there were nine European descent boy babies born on a given day and the mean birth weight is calculated. a.) State the random variable. Solution:

x = birth weight of boy babies (Note: the random variable is something you measure, and it is not the mean birth weight. Mean birth weight is calculated.)

b.) What is the mean of the sample mean? Solution: µx = µ = 3687.6 g c.) What is the standard deviation of the sample mean? Solution:

σ x = σ n = 410.5

9 = 410.5

3 ≈136.8 g

d.) What distribution is the sample mean distributed as? Solution:

Since the original random variable is distributed normally, then the sample mean is distributed normally.

e.) Find the probability that the mean weight of the nine boy babies born was less

than 3500.4 g. Solution:

You are looking for the P x < 3500.4( ) . You use the normalcdf command on the calculator. Remember to use the standard deviation you found in part c. However to reduce rounding error, type the division into the normalcdf command. You would have

P x < 3500.4( ) = normalcdf −1E99,3500.4,3687.6,410.5 ÷ 9( ) ≈ 0.086 There is an 8.6% chance that the mean birth weight of the nine boy babies born would be less than 3500.4 g. Since this is more than 5%, this is not unusual.

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f.) Find the probability that the mean weight of the nine babies born was less than 3452.5 g.

Solution: