1. What key characteristics does the central limit theorem tell us about a distr

Question

1. What key characteristics does the central limit theorem tell us about a distribution of means?

2. Please answer the following true or false questions.

The mean of a distribution of means is equal to the population mean the sample is drawn from. True or False ?

The variance of a distribution of means is always more than the variance of the population it is drawn from. True or False?

The shape of a distribution of means is always normal regardless of the population distribution when the sample size is more than 30. True or False?

3. A researcher wants to know if adults trained in a stress reduction method will respond differently after a traumatic event compared to the general population of adults (i.e., people not trained in the stress-reduction method). Stress responses in the general population are normally distributed with a mean of 40 and standard deviation of 10. A researcher tests the stress reduction method on a sample of 20 adults and finds they show a mean stress response of 30 after a traumatic event. Using the standard .05 cutoff, what should the researcher conclude about the effectiveness of the stress reduction method? Solve this problem explicitly using all five steps of hypothesis testing, and state your final conclusion in words.

Step 1: Restate question as a research hypothesis and a null hypothesis about populations.

Population 1 (1):

Population 2 (2):

Research hypothesis (H1):

Null hypothesis (H0):

Step 2: Determine the characteristics of the comparison distribution (i.e., its mean, variance, standard deviation, and shape).

Copy Sample Info Here:

Copy Population Info Here:

Mean: M =

Variance: 2M = 2 / N

SD: M = 2M

Shape:

Step 3: Determine the cutoff sample score (or Z score) on the comparison distribution at which the null hypothesis should be rejected.

Step 4: Determine the sample score on the comparison distribution.

Step 5: Decide whether to reject the null hypothesis.

State your conclusion in plain English. Give appropriate descriptive statistics (the sample mean, population mean, population standard deviation) and appropriate inferential statistics (the test you used, the score on the test, and the probability level). See the slides for examples.

4. Calculate the 95% confidence interval around the point estimate of the mean you solved for in problem 3. Solve this problem explicitly using the three steps we went over in class.

Step 1: Figure the sample standard error. (Hint: you found this in problem 3).

Step 2: Calculate interval width based on confidence level.

Step 3: Calculate the 95% confidence interval by adding and subtracting the width from the point estimate of the mean.

Extra Credit: If you wanted to conduct a Z-test like the one you conducted in problem 3, what information do you need to know about the population you are comparing your sample data to? Why might this information be hard to obtain?

Explanation / Answer

1. The central limit theorem tells us exactly what the shape of the distribution of means will be when we draw repeated samples from a given population. Specifically, as the sample sizes get larger, the distribution of means calculated from repeated sampling will approach normality. What makes the central limit theorem so remarkable is that this result holds no matter what shape the original population distribution may have been.

2.

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