Question 1

A newspaper reports that 26% of 1910 professionals said the biggest interview turnoff is that the applicant did not make an effort to learn about the job or company. The margin of error is given as $$\pm 3$$ percentage points. What important feature of the poll was omitted?

Accepted Answer

The confidence level

Question 2

A magazine poll of 500 adults found that 12% chose chocolate pie as their favorite, with a margin of error of $$\pm 5$$ percentage points. What does this margin of error mean?

Accepted Answer

The statement indicates that the interval 12\% $$\pm 5$$\% is likely to contain the true population percentage of people that prefer chocolate pie.

Question 3

A poll of 1000 adults found that 13\% chose chocolate pie, with a margin of error of $$\pm 3$$ percentage points. If the confidence level is 90\%, what is the value of $$\alpha$$?

Accepted Answer

$$\alpha = 0.10$$

Question 4

Given a poll of 2000 adults with 12\% choosing chocolate pie and a margin of error of $$\pm 4$$ percentage points, which confidence interval is wider: the 90\% or the 80\% confidence interval? Why?

Accepted Answer

A 90\% confidence interval must be wider than an 80\% confidence interval in order to be more confident that it captures the true value of the population proportion.

Question 5

Find the critical value $$z_{\alpha/2}$$ that corresponds to a confidence level of 85\%.

Accepted Answer

$$z_{\alpha/2} = 1.44$$

Question 6

Find the critical value $$z_{\alpha/2}$$ that corresponds to a confidence level of 94\%.

Accepted Answer

$$z_{\alpha/2} = 1.88$$

Question 7

Express the confidence interval $$0.666 \u003c p \u003c 0.888 in $$the form $$\hat{p} \pm E.$$

Accepted Answer

$$\hat{p} \pm E = 0.777 \pm 0.111$$

Question 8

Express the confidence interval $$(0.040, 0.104) in $$the form $$\hat{p} - E \u003c p \u003c \hat{p} + E.$$

Accepted Answer

$$0.072 - 0.032 \u003c p \u003c 0.072 + 0.032$$

Question 9

A research institute poll asked 984 respondents if they felt vulnerable to identity theft. 595 said "yes." Use a 95\% confidence level. What is the best point estimate of the population proportion $$p$$?

Accepted Answer

$$0.605$$

Question 10

In a clinical trial, among 4501 patients treated with a drug, 100 developed nausea. Construct a 95\% confidence interval for the proportion of adverse reactions. What is the margin of error $$E ($$rounded to three decimal places)?

Accepted Answer

$$E = 0.004$$

Question 11

A random sample of 862 births in a state included 438 boys. Construct a 95\% confidence interval estimate of the proportion of boys in all births. Is there strong evidence against the belief that the proportion of boys is 0.511?

Accepted Answer

There is not strong evidence against 0.511 as the value of the proportion of boys in all births because 0.511 is contained within the 95\% confidence interval.

Question 12

A genetic experiment with peas resulted in one sample of offspring that consisted of 421 green peas and 159 yellow peas. Construct a 95\% confidence interval to estimate the proportion of yellow peas. Does the result contradict the expectation that 25\% of the offspring peas would be yellow?

Accepted Answer

No, the confidence interval includes 0.25, so the true percentage could easily equal 25\%.

Question 13

In a tennis tournament, women playing singles matches used challenges on 131 calls made by line judges. Among those challenges, 31 were found to be successful. Construct a 95\% confidence interval for the percentage of successful challenges. If the 95\% confidence interval for men is $$16.4\% \u003c p \u003c 31.0\%$$, does it appear that either gender is more successful than the other?

Accepted Answer

Since the two confidence intervals overlap, neither gender appears to be substantially more successful in their challenges.

Question 14

A New York Times article reported that a survey conducted in 2014 included 36,000 adults, with 3.6\% of them being regular users of e-cigarettes. If a confidence level of 95\% and a margin of error of 3 percentage points are wanted, does the use of the result from the 2014 survey have much of an effect on the sample size required for a new survey?

Accepted Answer

Yes, using the result from the 2014 survey dramatically reduces the sample size.

Question 15

You are the operations manager for an airline and want to be 90\% confident that the sample percentage is within 3.5 percentage points of the true population percentage of passengers who prefer aisle seats. If nothing is known about the percentage, what is the minimum sample size required?

Accepted Answer

$$n = 653$$

Question 16

A sample of 115 body temperatures of healthy adults has a mean of $$98.6^\circF $$and a standard deviation of $$0.7^\circF. $$Construct a 99\% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of $$98.6^\circF as $$the mean body temperature?

Accepted Answer

The confidence interval does not contain $$98.6^\circF$$, suggesting that the mean body temperature is lower than $$98.6^\circF.$$

Question 17

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 15 subjects had a mean wake time of 105.0 min. After treatment, the mean wake time was 101.2 min with a standard deviation of 20 min. Construct a 95\% confidence interval for the mean wake time after treatment. What does the result suggest about the mean wake time of 105.0 min before the treatment?

Accepted Answer

The confidence interval includes the mean wake time of 105.0 min before the treatment, so the means before and after the treatment could be the same. This result suggests that the drug treatment does not have an effect.

Estimating Parameters and Determining Sample Sizes: Confidence Intervals and Margin of Error

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