(a) (i) Use StatCrunch to simulate 1000 random samples of size n = 10 from this population.

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1 Chapter 8 Sampling Distribution Ch 8.1 Distribution of Sample Mean Objective A : Shape, Center, and Spread of the Distributions of A1. Sampling Distributions of Mean A1.1 Sampling Distribution of the Sample Mean: Normal Population Eample 1: IQ is a measurement of intelligence derived from the Stanford Binet IQ test. Scores on this test are normally distributed with a mean score of 100 and a standard deviation of 15. (a) (i) Use StatCrunch to simulate 1000 random samples of size n = 10 from this population. StatCrunch -->StatCrunch Website -->Open StatCrunch --> Data --> Simulate --> Normal Click Options and Save. Eplain the representation of entries in row 1 of the StatCrunch spreadsheet. 1

2 (ii) Use StatCrunch to calculate the sample mean for each random sample of size n = 10. Stat --> Summary Stats --> Rows ---> Select all Normal1 to Normal10 for Select column(s) --> Select Mean for Statistics --> Check Store in data table --> Compute --> Options ---> Save. Eplain the representation of entries in column 11 of the StatCrunch spreadsheet. 2

3 (iii) Use StatCrunch to draw a histogram for the 1000 sample means. What is the sampling distribution of the sample means of sample size n = 10? Let's use a lower class limit of the first class of 65 and a class width of 5. Graph --> Histogram --> Select Row Mean for Select column(s) --> Input Start at: 65 and Width: 5 for Bins ---> Compute ---> Options ---> Save. Sketch the histogram and comment on the shape of the distribution. 3

4 (b) Use StatCrunch to find the mean and standard deviation of the sampling distribution of the 1000 sample means? Stat ---> Summary Stats ----> Columns ---> Select Row Mean ---> Select Mean and Std. dev. for Statistics ---> Compute ---> Options ---> Save. Write down the results. µ = = (c) Repeat part (a) and (b) with size n = 40. Sketch the histogram and comment on the shape of the distribution. Write down the results for µ = and = 4

5 A1. Sampling Distributions of Mean A1.2 Sampling Distribution of the Sample Mean: Nonnormal Population Eample 1: The waiting time in line can be modeled by an eponential distribution which is similar to skewed to the right with a mean of 5 minutes and a standard deviation of 5 minutes. (a) Repeat Eample 1 of A1.1 or part (a) but using eponential distribution instead of normal distribution. StatCrunch -->StatCrunch Website -->Open StatCrunch --> Data --> Simulate --> Eponential, then follow the steps given in (i) to (iii) to construct a histogram. If you are not sure about the start at value and the class width, just leave them blank and StatCrunch will figure it out based on the 1000 sample means. Sketch the histogram and comment on the shape of the distribution. (b) Repeat Eample 1 of A1.1 of part (b) using the sample means obtained from A1.2 of part (a). Write down the results. µ = = (c) Repeat part (a) and (b) with size n = 40. Sketch the histogram and comment on the shape of the distribution. Write down the results for µ = and = 5

6 A2. Central Limit Theorem A. If the population distribution of is normally distributed, the sampling distribution of is normally distributed regardless of the sample size n. If the population distribution is not normally distributed, the sampling distribution of is guaranteed to be normally distributed if 30 n. Use the distribution shapes obtained from the two simulation Eample 1 results of objective A1 to verify the statement A of the Central Limit Theorem. B. Mean/standard deviation of a sampling distribution of vs mean/standard deviation of a population distribution of. The mean and standard deviation of population distribution are µ and respectively. The mean of the sampling distribution of is µ where µ = µ. The standard deviation ofthe sampling distribution of is where = n. Use the µ and obtained from the two simulation Eample 1 of objective A1 to verify the statement B of the Central Limit Theorem. Eample 1 : Determine µ and from the given parameters of the population and the sample size. µ = 27, = 6, n = 15 Eample 2 : A simple random sample is obtained from a population with µ = 64 and = 18. (a) If the population distribution is skewed to the right, what condition must be applied in order to guarantee the sampling distribution of is normally distributed? 6

7 (b) If the sample size is n = 9, what must be true regarding the distribution of the population in order to guarantee the sampling distribution of to be normally distributed? Objective B : Finding Probability of that is Normally Distributed Standardize to Z Recall : Standardize to Z : Now : Standardize to Z : Z Z = = µ µ Eample 1 : A simple random sample of size n = 36 is obtained from a population mean µ = 64 and population standard deviation = 18. (a) Describe the sampling distribution. (b) Find µ and. (c) Use StatCrunch to find P< ( 62.6)? 7

8 Eample 2 : The upper leg of 20 to 29 year old males is normally distributed with a mean length of 43.7cm and a standard deviation of 4.2cm. (a) Use StatCrunch to the probability that a random sample of 12 males who are 20 to 29 years old results in a mean upper leg length that is between 42cm and 48cm? (b) A random sample of 15 males who are 20 to 29 years old results in a mean upper leg length greater than 46 cm. Do you find the result unusual? Why? 8

9 Ch 8.2 Distribution of the Sample Proportion Objective A : Shape, Center and Spread of the Distribution of. Distribution of the Sample Proportions - Eplain what is a distribution. A. Sampling distribution of sample proportion, where p ˆ =. n The shape of the sampling distribution of is approimately normally provided by, np(1 p) 10 or npq 10 where q = 1 p. B. Finding the mean and standard deviation of µ p, p ˆ = = p(1 p) n Objective B : Finding Probability of that is Normally Distributed Standardize to Z µ Z = where µ p ˆ = p and p(1 p) = n provided is approimately normally distributed. Eample 1: A nationwide study indicated that 80% of college students who use a cell phone, send and receive tet messages on their phone. A simple random sample of n = 200 college students using a cell phone is obtained. (a) Describe sampling distribution of. (b) If 154 college students in the sample send and receive tet messages on the cell phone, find, µ, and (c) Use StatCrunch to find what is the probability that 154 or fewer college students in the sample send and receive tet messages on the cell phone? Is this unusual? 9

10 Eample 2: According to creditcard.com, 29% of adults do not own a credit card. (a) Suppose a random sample of 500 adults is asked, "Do you own a credit card?" Describe the sampling distribution of, the proportion of adults who own a credit card. (b) Use StatCrunch to find the probability that in a random sample of 500 adults between 25% and 30% do not own a credit card? (c) Would it be unusual for a random sample of 500 adults to result in 125 or fewer who do not own a credit card? Why? 10