Exploring the Sampling Distribution of the Sample MeanObjective: To discover the properties of the sampling distribution of the sample mean and apply the Central Limit Theorem in problem-solving situations.1.Go to www.rossmanchance.com/applets2.Open the Sampling Pennies applet located under Classics3.If the page appears blank, you may need to click the link to open the javascript versionPart 1: Sampling Pennies1.Under population data select Pennies which gives the age, and year minted of a random sample of 1000 pennies. Consider the 1000 pennies the population.2.Select the variable “Year.” Describe the population distribution using SOCS (shape, outliers, center, spread)3.Click on the “Show Sampling Options” box in the upper right hand corner.4.Set the number of samples to 1 and the sample size to 10. Click on Draw Samples. You will see the sample data and the most recent sample in the graph below the sample data. The sample mean of that sample will be computed and you will see it plotted on the graph on the right. 5.Click Draw Samples a few more times. For each new sample another sample mean is computed and plotted in the graph on the right. 6.Make a prediction: What do you think the shape of the sampling distribution will be?7.Now, change the number of samples to 1000 and click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.8.Click on the Reset button. Change the sample size to 30 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.9.Click on the Reset button. Change the sample size to 50 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.10.Click on the Reset button. Change the sample size to 100 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.11.How did the sampling distribution of the sample means change as the sample size increased?Part 2: Sampling Change1.Under the population data select Change which gives the amount of change in the pockets of a random sample of 1000 college students. Consider the 1000 observations the population.2.Describe the population distribution using SOCS (shape, outliers, center, spread).3.Make a prediction: What do you think the shape of the sampling distribution will be? 4.Click on the “Show Sampling Options” box in the upper right hand corner. Set the number of samples to 1000 and the sample size to 10. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.5.Click on the Reset button. Change the sample size to 30 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.6.Click on the Reset button. Change the sample size to 50 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.7.Click on the Reset button. Change the sample size to 100 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.8.How did the sampling distribution of the sample means change as the sample size increased?Part 3: Sampling Stars1.Under the population data select Stars which gives the number of stars visible on 100 randomly selected nights. Consider the 100 observations the population.2.Select the variable “stars.” Describe the population distribution using SOCS (shape, outliers, center, spread)3.Make a prediction: What do you think the shape of the sampling distribution will be?4.Click on the “Show Sampling Options” box in the upper right hand corner. Set the number of samples to 1000 and the sample size to 10. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.5.Click on the Reset button. Change the sample size to 30 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.6.Click on the Reset button. Change the sample size to 50 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.7.Click on the Reset button. Change the sample size to 100 and the number of samples to 1000. Click on Draw Samples. Describe the distribution of the sample means using SOCS. Insert a screenshot of the sampling distribution below.8.How did the sampling distribution of the sample means change as the sample size increased?Part 4: Putting it all togetherThe Central Limit Theorem tells us that no matter the shape of the population distribution the sampling distribution will be approximately normal as long as the 4 conditions below have been met.1.The samples have been obtained randomly2.The samples are independent3.The sample size is less than 10% of the population4.The sample size is large enoughPlease note: for the sampling distribution of the sample mean ‘large enough’ depends upon the shape of the population distribution. For population distributions that are approximately normal, very small sample sizes are ‘large enough.’ However, for very skewed distributions the sample size must be much larger than the size of 30 that your text recommends.Formulas for the mean and standard deviation of the sampling distribution of the sample meanMean: Standard Deviation: 1.Complete the table below that summarizes the result of each of your investigations and compute the mean and standard deviation for the sampling distributions for samples of size 100. Pennies Change StarsShape of Population Distribution Population Mean Population SD Shape n = 10 Shape n = 30 Shape n = 50 Shape n = 100 Mean n = 100 SD n = 100 n = 100, n = 100, Part 5: Applying what you’ve learned1.For a sample size of 100, find the probability of obtaining a sample mean amount of change less than 0.45.2.For a sample size of 100, find the probability of obtaining a sample mean amount of change greater than 0.53.3.For a sample size of 100, find the probability of obtaining a sample mean amount of change between 0.51 and 0.57.4.For a sample size of 100, find the 80th percentile of the sample mean amount of change.For a sample size of 100, what would be unusual sample mean amounts of change? 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