Question

A simple random sample of 1100 males aged 12 to 17 in the United States were asked whether they played massive multiplayer online role-playing games (MMORPGs); 775 said that they did. a. We want to use this information to construct a 95% confidence interval to estimate the proportion of all U.S. males aged 12 to 17 who play MMORPGs.

State the parameter our confidence interval will estimate (in context).

b. State the value of our point estimate (i.e., the statistic, ). Round to four decimal places.

c. Identify the conditions that must be met to use this procedure and explain how you know that each one has been satisfied.

d. Find the appropriate critical value () and the standard error of the sample proportion (). SHOW YOUR WORK! Round the standard error to four decimal places.

e. Use the formula shown in your notes to get the 95% confidence interval by hand. SHOW YOUR WORK! Round to four decimal places.

f. Interpret the confidence interval constructed in part (e) in the context of the problem.

g. Interpret the confidence level in the context of the problem.

h. Suppose you wanted to estimate the proportion of 12-to-17 year-old males who play MMORPG’s with 95% confidence to have a margin of error within ± 2%. Calculate how large a sample you would need. Use the found in (b). SHOW YOUR WORK! Remember to round your final answer up to the nearest whole number.

i. If you wanted to have a margin of error of ±2% with 99% confidence, would your sample have to be larger, smaller, or the same size as the sample in part (h)? Explain.

j. Suppose MMORPG.com claims that 65% of all U.S. males aged 12-17 play massive multiplayer online role-playing games (MMORPGs). Does your 95% confidence interval support this claim?

k. What is the name of the significance test that can we perform to test the claim made?

l. What hypotheses would we use if we wished to conduct a two-sided test?

m. The only condition that changed from earlier is the Success/Failure Condition. You must now verify the expected number of successes and failures using our null hypothesis value.

n. Now we can proceed with the calculations of the standard deviation (). SHOW YOUR WORK! Round to four decimal places. Note: Remember to use and in the formula.

o. Calculate the z-score. SHOW YOUR WORK!

p. Use your z-score to calculate the p-value. SHOW YOUR WORK! Hint: Remember to double the p-value if you start by calculating a one-sided probability.

q. Interpret the p-value in context.

r. What decision would you draw based on the size of the p-value?

s. Are our confidence interval and significance test results in agreement?

Answer 1

a.

The parameter that our confidence interval will estimate is the proportion of all U.S. males aged 12 to 17 who play MMORPGs, i.e., population proportion, p.

b.

Point estimate =Sample proportion, =x/n =775/1100 =0.7045

c.

1) The sample must be drawn randomly which is given in this case.

2) The sample values have to be independent of each other. Since the sample is drawn randomly, we can be sure that they are independent of each other.

3) If the sample is drawn without replacement (which must be the case here), the sample size, n, should not be more than 10% of the population. We have n =1100 and we can be sure enough that our population of males aged 12 to 17 in the United States are more than 11000.

4) The sample size must be large enough for the Central Limit Theorem to hold good so that we can use normal distribution. If n > 30, it is considered large enough. Here, n =1100 which is a large sample.

d.

Critical value of Z at 95% confidence level for a two-tailed case is: Z-critical =1.96

Standard Error of sample proportion, SE() =​​​​​​) = =0.0138

e.

95% confidence interval for the population proportion, p is:

p =[Z-critical*SE(​​​​​​)] =0.7045(1.96*0.0138) =[0.6775, 0.7315]

f.

Interpretation of confidence interval:

We are 95% confident that the interval [0.6775, 0.7315] contains the true proportion of all U.S. males aged 12 to 17 who play MMORPGs.

g.

Interpretation of 95% confidence level:

If we drew many random samples of size 1100 and constructed 95% confidence intervals, then we would expect 95% of such intervals contain the true proportion of all US males aged 12 to 17 who play MMORPGs and 5% of such intervals do not contain it. This 5% is called the significance level.