Central limit theorem ti 83

Statistical Methods I

Chapter 5: Normal Probability Distributions

Section 5.1: Intro to Normal Distributions and the Standard Normal Distributions Objectives:

 Normal Distribution Properties

 Use z-scores to Calculate Area Under the Standard Normal Curve (using StatCrunch or Calculator)

 Discuss Unusual Values

In this section we will revisit histograms which can be estimated with normal (symmetric, bell-shaped) curves. From Test 1 remember that normal curves have z-scores (for any data value) and areas under the curve (one way: Empirical Rule). Now we will use these normal curves to find probabilities (areas) and z-scores for any data value. Why do we need to study this? Eventually we will use these probabilities and z-scores to make decisions. By using the normal distribution curve, we are treating the data as a continuous random variable that has its own continuous probability distribution. (Remember that any probability distribution has two properties: all probabilities are between 0 and 1 and the sum of the probabilities is 1.) **Probabilities = Areas under the curve**

Ex: Consider the normal distribution curves below. Which normal curve has the greatest mean? Which normal curve has the greatest standard deviation?

Note: Every normal distribution can be transformed into the Standard Normal Distribution (the distribution for z- scores). This means we will use the z-score formula to transform any data value into a “measure of position” with the formula:

data value mean

standard deviation z

 

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**All probability calculations will be done with either StatCrunch or the TI 83/84 calculator. You do NOT need to learn how to read the Standard Normal Table.**

**Also < and  are treated the same as well as > and  for any continuous probability distribution.**

Ex: Confirm that the area to the left of z = 1.15 is 0.8749. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, enter inequality symbol and z-score, Compute TI-83/84: 2nd  VARS normalcdf( -1000000000  Comma  1.15 Comma  0  Comma 1 enter P(z  1.15) = 0.8749 Ex: Confirm that the cumulative area that corresponds to z = -0.24 is 0.4052. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, enter inequality symbol and z-score, Compute TI-83/84: 2nd  VARS normalcdf( -1000000000  Comma  -0.24 Comma 0  Comma 1 enter P(z  -0.24) = 0.4052

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Ex: Find the area to right of each z-score. Hint: Use the fact that the total area (probability) is 1. **Label the z-score and the area.** a) b)

P(z  1.15) = _________________ P(z  -0.24) = _________________ Ex: Find the shaded area. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, enter inequality symbol and z-score, Compute TI-83/84: 2nd  VARS normalcdf( -2.3  Comma  1000000000 Comma 0  Comma 1 enter

P(z  -2.3) = _________________ Ex: Find the shaded area. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Between, enter z-scores, Compute TI-83/84: 2nd  VARS normalcdf( -1.5  Comma  1.25 Comma 0  Comma 1 enter

P(-1.5  z  1.25) = _________________ Ex: The SAT is an exam used by colleges and universities to evaluate undergraduate applicants. The test scores are normally distributed. In a recent year, the mean ( ) test score was 1498 and the standard deviation ( ) was 316. The

test scores of four students selected at random are 1920, 1240, 2350, and 1390. Sketch this distribution, find the z-scores for each value, and determine whether any of the values are unusual when compared to the mean and standard deviation.

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Section 5.2: Normal Probability Distributions: Finding Probabilities Objectives:

 Sketch Normal Distribution along with Standard Normal Distribution

 Find z-scores to calculate Area Under the Normal Curve (using StatCrunch or Calculator)

In this section you will get the chance to apply the probabilities (areas) from the Standard Normal Distribution to real- life situations. Consider the last example concerning the SAT: Ex: The SAT is an exam used by colleges and universities to evaluate undergraduate applicants. The test scores are normally distributed. In a recent year, the mean ( ) test score was 1498 and the standard deviation ( ) was 316. The

test scores of four students selected at random are 1920, 1240, 2350, and 1390. Sketch this distribution and find the area to the right (probability above) for each score. Are any of these events unusual? **Remember: Any probability that is 0.05 or below means that an event is considered unusual… Student 1: score = 1920 Student 2: score = 1240 z-score = _______________ z-score = _______________ P(x > 1920) = P(z > _____ ) = ________________ P(x > 1240) = P(z > _____ ) = ________________ Unusual Event? ________________ Unusual Event? ________________ Student 3: score = 2350 Student 4: score = 1390 z-score = _______________ z-score = _______________ P(x > 2350) = P(z > _____ ) = ________________ P(x > 1390) = P(z > _____ ) = ________________ Unusual Event? ________________ Unusual Event? ________________ StatCrunch: Stat menu, Calculators, Normal, Standard, enter mean & standard deviation, enter inequality symbol and x value, Compute TI-83/84: 2nd  VARS normalcdf( x value  Comma  1000000000 Comma mean  Comma standard deviation enter

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Ex: A survey indicates that people keep their cell phone an average of 1.5 years before buying a new one. The population standard deviation is 0.25 year. Assume that the lengths of time people keep their phone are normally distributed and are represented by the variable x. A cell phone user is selected at random. Find each probability and show a sketch for each one. **Label the z-score and the area.**

a. Find the probability that the user will keep his/her current phone for less than 1 year before buying a new one.

b. Find the probability that the user will keep his/her current phone for more than 1.8 years before buying a new

one.

c. Find the probability that the user will keep his/her current phone between 1 year and 1.8 years before buying a

new one.

d. Out of 1000 people, about how many would you expect to keep his/her current phone between 1 year and 1.8

years before buying a new one? Round to the nearest whole number.

e. Since the mean is 1.5 years and the standard deviation is 0.25 year, do we expect someone to keep his/her

current phone for more than 1.8 years? Explain what your answer means.

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5.3: Normal Distributions: Finding Values Objectives:

 Find a z-score Given the Area Under the Normal Curve

 Find the z-score for a given Percentile

 Transform a z-score to an x-value

 Find the x-value that separates the Top or Bottom Percentage of the Area

In the previous section we calculated probabilities based on given x values. What if we start with a probability, can we find the corresponding x value (or z-score)? Ex: Find the z-score that corresponds to a cumulative area of 0.3632. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, choose symbol, enter probability, Compute TI-83/84: 2nd  VARS invNorm( area to left  Comma  0  Comma 1 enter

Ex: Find a z-score that has 10.75% of the distribution’s area to its right. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, choose symbol, enter probability, Compute TI-83/84: 2nd  VARS invNorm( area to left  Comma  0  Comma 1 enter

Ex: Find the z-score that corresponds to the 5th percentile: P5. **Label the sketch and estimate its value first.** **Label the z-score and the area.**

Ex: Find the z-score that corresponds to the 93rd percentile: P93. **Sketch this distribution and estimate the z-score first.** **Label the z-score and the area.**

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Ex: Find the z-score described in the graph. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Between, enter probability, Compute TI-83/84: 2nd  VARS invNorm( area to left  Comma  0  Comma 1 enter

Ex: Find the z-score for which 84% of the distribution’s area lies between –z and z. **Sketch this distribution and estimate the z-scores first.** **Label the z-score and the area.** Ex: Find the z-score described in the graph. **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, enter probability, Compute TI-83/84: 2nd  VARS invNorm( area to left  Comma  0  Comma 1 enter

Remember that we can transform any x value into a z-score. How will we do the reverse and transform any z-score into an x value?

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Ex: A veterinarian records the weights of cats treated at a clinic. The weights are normally distributed, with a mean of 9 pounds and a standard deviation of 2 pounds. Find the weights x corresponding to z-scores of 1.96, -0.44, and 0. Interpret each result and show these on a sketch below.

Now let’s find an x value given a probability… Ex: Scores for the California Peace Officer Standards and Training test are normally distributed, with a mean of 50 and a population standard deviation of 10. An agency will only hire applicants with scores in the top 10%. What is the lowest score an applicant can earn and still be eligible to be hired by the agency? **Label the z-score and the area.** StatCrunch: Stat menu, Calculators, Normal, Standard, enter mean & st dev, choose symbol, enter probability, Compute TI-83/84: 2nd  VARS invNorm( area to left  Comma  mean  Comma standard deviation enter

Ex: A researcher tests the braking distances of several cars. The braking distance from 60 miles per hour to a complete stop on dry pavement is measured in feet. The breaking distances of a sample of cars are normally distributed, with a mean of 129 feet and a population standard deviation of 5.18 feet. What is the longest braking distance one of these cars could have and still be in the bottom 1%? **Sketch & label the z- score and the area.**

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Ex: The weights of bags of baby carrots are normally distributed, with a mean of 32 ounces and a population standard deviation of 0.36 ounce. Bags in the upper 4.5% are too heavy and must be repackaged. What is the most a bag of baby carrots can weigh and not need to be repackaged? **Sketch & label the z-score and the area.**

5.4: Sampling Distributions and the Central Limit Theorem Objectives:

 Sampling Distribution Properties

 Use the Central Limit Theorem

 Calculate the Probability of a Sample Mean

 Compare Probabilities for x and x

Now that we have studied how a continuous random variable behaves, we will now focus on the relationship between a population mean and the mean of samples taken from a population. Remember that different samples have different means!! Def: A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of sample means. Every sample statistic has a sampling distribution. For the population below, 5 samples of size n have been taken (with replacement) from the population and each sample has its own sample mean. If we take an infinite number of these samples and look at each sample mean, the combined result of all of the sample means will be the sampling distribution.

Now… who wants to take an infinite number of samples from a population and combine those results?

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The good news is that we do not have to take an infinite number of samples from a population and combine those results… We can use the following properties & the Central Limit Theorem to describe the new sampling distribution.

The Central Limit Theorem The Central Limit Theorem is the foundation for inferential statistics and it describes the relationship between the sampling distribution of sample means and the population from which the samples are taken from. This theorem will allow us to use sample statistics to make inferences about the population mean. **Remember that if we EXPECT a particular mean, then we find it UNUSUAL when sample results are FAR AWAY from the mean…

**The distribution of sample means has the same mean as the population mean but its standard deviation is less than the standard deviation of the population… So both distributions have the same center but the standard deviation of the sampling distribution is less spread out… And as n increases, the sampling distribution gets even smaller and there is less spread around the mean.

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Ex: Cell phone bills for residents of a city have a mean of $47 and a population standard deviation of $9. Random samples of 100 cell phones bills are drawn from the population, and the mean of each sample is determined. Find the mean and standard deviation of the sampling distribution of sample means. Compare the sketches below.

Whether we are working with a population or a sampling distribution of sample means, we can calculate z-scores and probabilities (areas) in the same way… The only difference is that we are required to calculate a z-score based on the NEW standard deviation called standard error.

value mean

standard error n

x z



   

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Ex: The mean annual salary for flight attendants is about $65,700. A random sample of 48 flight attendants is selected from this population. Assume  = $14,500. Sketch the sampling distribution, state the mean, the standard error, and the corresponding z-score for $63,400. What is the probability that the mean annual salary of the sample is less than $63,400? StatCrunch: Stat menu, Calculators, Normal, Standard, enter mean & NEW standard deviation, enter x value and inequality symbol, Compute TI-83/84: 2nd  VARS normalcdf( -1000000000  Comma  x value  Comma mean  Comma NEW standard deviation enter Ex: A machine is set to fill milk containers with a mean of 64 ounces and a population standard deviation of 0.11 ounce. A random sample of 40 containers has a mean of 64.05 ounces. Does the machine need to be reset? Explain.

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Ex: The weights of ice cream cartons produced by a manufacturer are normally distributed with a mean weight of 10 ounces and population standard deviation of 0.5 ounce.

a. What is the probability that a randomly selected carton has a weight greater than 10.21 ounces?

StatCrunch: Stat menu, Calculators, Normal, Standard, enter mean & standard deviation, enter inequality symbol and x value, Compute

TI-83/84: 2nd  VARS normalcdf( x value  Comma  1000000000 Comma mean  Comma standard deviation enter

b. You randomly select 25 cartons. What is the probability that their mean weight is greater than 10.21 ounces?

StatCrunch: Stat menu, Calculators, Normal, Standard, enter mean & NEW standard deviation, enter inequality symbol and x value, Compute

TI-83/84: 2nd  VARS normalcdf( x value  Comma  1000000000 Comma mean  Comma NEW standard deviation enter

c. Compare these two answers.

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Ex: A manufacturer claims that the life span of its tires is 50,000 miles. You work for a consumer protection agency and you are testing this manufacturer’s tires. Assume the life spans of the tires are normally distributed. You select 100 tires at random and test them. The mean life span is 49,721 miles. Assume  = 800 miles.

a. Assuming the manufacturer’s claim is correct, what is the probability that the mean of the sample is 49,721

miles or less?

b. Using your answer from part a, what do you think of the manufacturer’s claim?

c. Would it be unusual to have an individual tire with a life span of 49,721 miles or less? Why or why not?

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Often the population standard deviation is unknown. But, in cases like this we can still find a “score” that corresponds to an area using the t-distribution and the sample standard deviation.

Ex: Confirm that the area to the left of t = 1.15 is 0.8615 when n = 11. **Label the t-score and the area.** StatCrunch: Stat menu, Calculators, T, enter degrees of freedom, inequality symbol and t-score, Compute TI-83/84: 2nd  VARS tcdf( -1000000000  Comma  1.15 Comma  degrees of freedom  enter P(t  1.15) = 0.8615 **Notice this answer is CLOSE to the previous z-score area of 0.8749…

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Ex: Confirm that the cumulative area that corresponds to t = -0.24 is 0.4062 when n = 24. **Label the t-score and the area.** StatCrunch: Stat menu, Calculators, T, enter degrees of freedom, inequality symbol and t-score, Compute TI-83/84: 2nd  VARS tcdf( -1000000000  Comma  -0.24 Comma  degrees of freedom  enter P(t  -0.24) = 0.4062 **Notice this answer is CLOSE to the previous z-score area of 0.4052… Ex: Find the shaded area to the right of t = -2.3 when n = 45. **Label the t-score and the area.** StatCrunch: Stat menu, Calculators, T, enter degrees of freedom, inequality symbol and t-score, Compute TI-83/84: 2nd  VARS tcdf( -2.3  Comma  1000000000 Comma  degrees of freedom  enter Ex: Find the shaded area between t = -1.5 and t = 1.25 when n = 93. **Label the t-score and the area.** StatCrunch: Stat menu, Calculators, T, enter degrees of freedom, inequality symbol and t-score, Compute TI-83/84: 2nd  VARS normalcdf( -1.5  Comma  1.25 Comma  degrees of freedom  enter Ex: Find the t-score that corresponds to a cumulative area of 0.3632 when n = 67. **Label the t-score and the area.** StatCrunch: Stat menu, Calculators, T, enter degrees of freedom, inequality symbol and area, Compute TI-83/84: 2nd  VARS invT( area to left  Comma  degrees of freedom  enter Ex: Find a t-score that has 10.75% of the distribution’s area to its right when n = 35. **Label the t-score and the area.** StatCrunch: Stat menu, Calculators, T, enter degrees of freedom, inequality symbol and area, Compute TI-83/84: 2nd  VARS invT( area to left  Comma  degrees of freedom  enter