1.           Suppose a professor gives an exam to a class of 40 students and the scores are as follows.

Find each of the following:

Mean:

Median:

Standard deviation:

Z-score for a student who originally scored 54 on the exam:

Create a histogram of the scores and comment on the shape of the distribution.

2.           Suppose the professor decides to adjust scores by adding 25 points to each student’s score.

a.           Find each of the following after the adjustment:

Mean:

Median:

Standard deviation:

Z-score for a student who originally scored 54 on the exam:

Create a histogram of the adjusted scores and compare it to the histogram created in #1.

1.

Load the Sampling Distribution Bell Shaped Applet from www.pearsonhighered.com/sullivanstats or from StatCrunch.  Set the applet so that the          population is bell-shaped.  Decide on a mean and standard deviation and make a note of them here.

Obtain 1,000 random samples of size n = 5, n = 10, and n = 20.

Describe the distribution of the sample means based on the results of the applet.  That is, describe the center (mean), spread (standard deviation), and shape of the distribution.

d.           Compare the distributions for the various sample sizes.

e            What role, if any, does the sample size play in the sampling distribution of the sample mean?

2.

Load the Sampling Distribution Uniform Applet from www.pearsonhighered.com/sullivanstats or from StatCrunch. Set the applet so that the population is uniform.

Obtain 1,000 random samples of size n = 2, n = 5, n = 10, n = 20, and n = 30.

You may stop generating random samples once the distribution of the sample mean is approximately normal.

Record the mean and the standard deviation of the distribution of sample means for each sample size.

126     Sampling from Normal and Non-Normal Populations

d.           For which sample size is the distribution of sample means approximately normal?

e.           What is the mean of each distribution (regardless of shape)?

f.            What is the standard deviation of each distribution (regardless of shape)?

g.           How does the size of the sample affect the standard deviation of the sample means?

3.

Draw a skewed distribution on the applet by holding the left mouse button down and dragging the mouse over the distribution.

Obtain 1,000 random samples of size n = 2, n = 5, n = 10, n = 20, and n = 30.

You may stop generating random samples once the distribution of the sample mean is approximately normal.

Record the mean and the standard deviation of the distribution of sample means for each sample size.

For which sample size is the distribution of sample means approximately normal?

What is the mean of each distribution (regardless of shape)?

What is the standard deviation of each distribution (regardless of shape)?

How does the size of the sample affect the standard deviation of the sample means?

The Logic of Hypothesis Testing

In this scenario, we are testing

Ho : p = 0.5  versus H1 : p > 0.5

Load the Political Poll Applet that is located at www.pearsonhighered.com/sullivanstats to simulate surveying   500 registered voters, assuming that 50% of all registered voters are in favor of the policy and 50% are against   the policy. This is done because we always assume that the statement in the null hypothesis is true until we have evidence to the contrary. So we assume that we are sampling from a population where the proportion of registered voters who are in favor of the policy is 0.5.

1.           Explain how 500 coins might be used to simulate the survey results.

2. Simulate obtaining a simple random sample of 500 voters, assuming that p = 0.5, exactly one time.  In this simulation, we are flipping 500 coins where each coin represents a voter.  The probability of        obtaining a head (a voter in favor of the policy) is 0.5.

How many heads did you obtain? (Click on Run 1 to see the number of heads.)   That is, how many registered voters are in favor of the policy?

What is the sample proportion of voters in favor of the policy?

3.

Simulate obtaining a second simple random sample of 500 voters, assuming that p = 0.5, exactly one time.

In this sample, how many registered voters are in favor of the policy? (Click on Run 2 to determine the number of heads.)

What is the sample proportion of voters in favor of the policy?

4. The sample proportion obtained in part (3) is likely different from the sample proportion obtained in part (2).  Why?

160     The Logic of Hypothesis Testing

5.           Suppose that the polling firm actually conducts a survey of 500 registered voters and finds that 260 are in favor of the policy so that pˆ = = 0.52 .

Do you believe that this sample evidence suggests that the true percentage of registered voters in favor of the policy is greater than 0.5? Or, is it reasonable (that is, not all that unlikely) to obtain a sample of 260 voters in favor of the policy even though the population proportion of voters in favor of the policy is 0.5?

To help answer those questions, simulate conducting a survey of 500 registered voters many times and determine the proportion of times you observe 260 or more in favor of the policy. To do this, “Reset” the applet.  Click “1000 runs” to simulate obtaining a simple random sample of 500 registered voters   1000 different times and recording the number of individuals in favor of the policy (assuming that the population proportion is 0.5). In the cell "# of heads," be sure the drop-down menu is set to >=. In the  cell to the right of the drop-down menu, type 260 and press Enter. Look in the Count column.

How many of the1000 different surveys resulted in 260 or more voters in favor of the policy?

What proportion of the 1000 different simulations results in a sample proportion of 0.52 or higher even though the population proportion is 0.5?

Do you consider this convincing evidence that the proportion of voters in favor of the policy is greater than 0.5? Why or why not?

What is the shape of the distribution of sample proportions?  What is the center? Explain why the shape and center are reasonable.

What if the polling firm discussed in part (5) conducts a survey of 500 registered voters and finds that 270 are in favor of the policy so that pˆ = = 0.54 ?

Do you believe that this sample evidence suggests that the true proportion of registered voters in favor of the policy is greater than 0.5?

Reset the applet. Click "1000 runs" to simulate obtaining a simple random sample of 500 registered   voters 1000 different times and record the number of individuals in favor of the policy (assuming that the population proportion is 0.5). In the cell “# of heads,” be sure the drop-down menu is set to >=. In the cell to the right of the drop-down menu, type 270 and press Enter. Look in the Count column.

How many of the 1000 different simulations resulted in 270 or more voters in favor of the policy?

What proportion of surveys results in a sample proportion of 0.54 or higher even though the population proportion is 0.5?

Do you consider this convincing evidence that the proportion of voters in favor of the policy is greater than 0.5? Why or why not?

The Logic of Hypothesis Testing     161

7.           What if the polling firm discussed in part (5) conducts a survey of 500 registered voters and finds that

280 are in favor of the policy so that pˆ = = 0.560 ?

Do you believe that this sample evidence suggests that the true proportion of registered voters in favor of the policy is greater than 0.5?

Reset the applet. Click “1000 runs” to simulate obtaining a simple random sample of 500 registered   voters 1000 different times and record the number of individuals in favor of the policy (assuming that the population proportion is 0.5). In the cell “# of heads,” be sure the drop-down menu is set to >=. In the cell to the right of the drop-down menu, type 280 and press Enter. Look in the Count column.

How many of the 1000 different surveys resulted in 280 or more voters in favor of the policy?

What proportion of surveys results in a sample proportion of 0.56 or higher even though the population proportion is 0.5?

Do you consider this convincing evidence that the proportion of voters in favor of the policy is greater than 0.5? Why or why not?

Note: The proportion of simulations that resulted in a sample proportion as extreme, or more extreme, than 0.52 (for simulation from part 5), 0.54 (for simulation from part 6), or 0.56 (for simulation from  part 7) is called the P-value.

8.           Describe the sampling distribution of pˆ under the assumption the statement in the null hypothesis is true.

9.           Compute P (pˆ > 0.52) assuming Ho : p = 0.5 is true.  Compare this result to the simulation from part

(5).  Repeat this for pˆ = 0.54   and pˆ = 0.56 . Finally, conclude that either simulation or a normal model (when the requirements are satisfied) may be used to approximate P-values for hypotheses involving a population proportion.

Analyzing a Research Article I

The following table summarizes some of the results in the meta-analysis for smokers who used e- cigarettes with nicotine.