Baseball Players’ Salaries. We have access to data regarding the salaries of all

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Baseball Players’ Salaries. We have access to data regarding the salaries of all professional baseball players in 2012. That is, if we consider all professional baseball players in 2012 our subjects of interest, then we have information on every individual in the population. In this problem, we are going to examine how varying the sample size impacts the sampling distribution of the sample mean. We will be using an applet called StatKey to complete this problem. Start by opening a web browser on your computer and going to the following website: http://lock5stat.com/statkey/sampling_1_quant/sampling_1_quant.html (g) Suppose we drew many samples of size n = 10 and calculated the sample mean for each of these samples. i. What value would you expect to see for the mean of the sample means (in millions of dollars) of size n = 10? (Report to the nearest 3 decimal places.) ii. What value would you expect to see for the standard error of the sample means (in millions of dollars) of size n = 10? (Report to the nearest 3 decimal places.) (i) We now wish to see what happens to the sampling distribution of the sample mean if we increase our sample size to n = 50. To do this, click on Reset Plot at the top of the page. Click on the blank next to Choose samples of size n= and enter 50 and hit OK. Select three different samples of size n = 50 and record the three sample means. i. What is the value of the sample mean (in millions of dollars) from your first random sample of size n = 50? (Report to the nearest 3 decimal places.) ii. What is the value of the sample mean (in millions of dollars) from your second random sample of size n = 50? (Report to the nearest 3 decimal places.) iii. What is the value of the sample mean (in millions of dollars) from your third random sample of size n = 50? (Report to the nearest 3 decimal places.) iv. (Free Response) The three sample means calculated from samples of size n = 50 should be much closer to each other than the three sample means calculate from size n = 10 (parts (c) - (e) above). Why is this the case? (HINT: Think about the standard error of the sampling distribution of the sample mean when n = 10 compared to when n = 50.) (j) Suppose we drew many samples of size n = 50 and calculated the sample mean for each of these samples. i. How would the mean of the sample means from samples of size n = 50 compare to the mean of the sample means from samples of size n = 10? (Choose one) • The mean of the sample means based on samples of n = 50 would be larger than the mean of the sample means based on samples of size n = 10. • The mean of the sample means based on samples of n = 50 would be smaller than the mean of the sample means based on samples of size n = 10. • The mean of the sample means based on samples of n = 50 would be about the same as the mean of the sample means based on samples of size n = 10. • There is not enough information to answer this question. ii. How would the standard error of the sample means from samples of size n = 50 compare to the standard error of the sample means from samples of size n = 10? (Choose one) • The standard error of the sample means based on samples of n = 50 would be larger than the standard error of the sample means based on samples of size n = 10. • The standard error of the sample means based on samples of n = 50 would be smaller than the standard error of the sample means based on samples of size n = 10. 5 • The standard error of the sample means based on samples of n = 50 would be about the same as the standard error of the sample means based on samples of size n = 10. • There is not enough information to answer this question. (k) Take 2,000 more samples of size n = 50 from the population. i. (Free Response) Take a screen shot of the sampling distribution of the sample mean for n = 50 and upload it to Blackboard. The file must be a JPG, PNG, or PDF file. ii. Report the value of the mean of the 2,003 sample means to 3 decimal places. iii. Report the value of the standard error of the 2,003 sample means to 3 decimal places. iv. (Free Response) How does the shape of the distribution of sample means for samples of size n = 50 compare to the shape of the distribution of sample means for samples of size n = 10? Explain why this is the case.

Explanation / Answer

Answer to part g)

(i) The mean of samples with sample size 10 is 34.59

(ii) the standard error = 30.674/root(10) = 9.7000

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Answer to part (j)

(i) This time for a sample of size 50 , we get a mean of 45.934 , which is greater than the mean of Samples with size 10 , and infact this mean is more close to the population mean 43.024

Thus the first answer choice is correct, that mean for samples with size 50 is greater than mean for samples with size 10

(ii) The standard error = 28.95/root(50) = 4.094

The standard error for larger sample is way too small as compared to a sample of smaller size

Thus the second answer choice is correct

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