5. Central Limit Theorem Twenty students from a statistics class each collected

ID: 3048658 • Letter: 5

Question

5. Central Limit Theorem Twenty students from a statistics class each collected a random sample of times on how long it took students to get to class from their homes. All the sample sizes were 30. The resulting means are listed. Mean 27 24 14 29 37 23 26 21 30 Student Std. Dev. Student 3.7 4.6 2.4 Std. Dev 2.2 2.4 31 27 20 17 26 34 13 14 15 16 3.0 2.7 1.8 2.0 2.2 2.8 18 2.6 2.1 20 The students noticed that everyone had different answers. If you randomly sample over and over from any population, with the same sample size, will the results ever be the same? The students wondered whose results were right. How can they find out what the population mean and standard deviation are? Input the means into the computer and check to see if the distribution is normal Check the mean and standard deviation of the means. How do these values compare to the students' individual scores? a. b. c. d. e. Is the distribution of the means a sampling distribution? f. Check the sampling error for students 3, 7, and 14 g. Compare the standard deviation of the sample of the 20 means. Is that equal to the standard deviation from student 3 divided by the square of the sample size? How about for student 7, or 14?

Explanation / Answer

Part (a)

No, even if the sample sizes are the same, the chances of the values being the same are very remote. In fact, all values will be different. ANSWER

Part (b)

There is nothing like correct answer.

Population mean can estimated by the mean of the 20 sample means.

Similarly, population standard deviation can also be estimated using sample standard deviations, though not by taking the mean. It involves converting the sample standard deviations to sample variances and then taking the mean. ANSWER

Part (c)

To address this part, we first draw a meaningful frequency distribution as shown below:

Minimum value: 14 and maximum value: 37. So, we design a frequency distribution of width of 4. Means being continuous variable, we first fix the class limits and then convert them to class boundaries.

Class Limits

Class Boundaries

Tally Marks

Frequency

14 - 17

13.5 – 17.5

18 - 21

17.5 – 21.5

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22 - 25

22.5 – 25.5

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26 - 29

25.5 – 29.5

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30 - 33

29.5 – 33.5

34 - 37

33.5 – 37.5

Considering the number of values is only as small as 20, the frequency distribution can be considered a Normal Distribution. It is fairly symmetric with respect to the class (26 – 29).

DONE

Part (d)

Mean of sample means = 25.35 ANSWER 1

Standard Deviation of sample means = 5.63 ANSWER 2

Excel Calculations are shown below:

AVERAGE(J2:J21)

25.35

STDEVP(J2:J21)

5.632