8. Sampling Distributions & Confidence Intervals: Proportion

In Exercise 11, would it be unusual for the population proportion to be 72.5%? Explain.

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 220 U.S. adults ages 18–29, 65% said that they use Snapchat. The survey’s margin of error is ±7.9%. (Source: Pew Research Center)

Make a confidence interval for $p$ given the following values.

$x=314,n=500,C=99\%$

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?

b. Assume that 11% of consumers have a smartphone and plan to upgrade to a new model.

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.

In a survey of 912 U.S. adults in Generation Z (born after 1996), 383 said they are at least somewhat likely to consider an electric vehicle for their next vehicle purchase. (Adapted from Pew Research Center)

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Touch Therapy When she was 9 years of age, Emily Rosa did a science fair experiment in which she tested professional touch therapists to see if they could sense her energy field. She flipped a coin to select either her right hand or her left hand, and then she asked the therapists to identify the selected hand by placing their hand just under Emily’s hand without seeing it and without touching it. Among 280 trials, the touch therapists were correct 123 times (based on data in “A Close Look at Therapeutic Touch,” Journal of the American Medical Association, Vol. 279, No. 13).

c. Using Emily’s sample results, construct a 99% confidence interval estimate of the proportion of correct responses made by touch therapists.

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Job Interviews In a Harris poll of 514 human resource professionals, 90% said that the appearance of a job applicant is most important for a good first impression.

a. Among the 514 human resource professionals who were surveyed, how many of them said that the appearance of a job applicant is most important for a good first impression?

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

If a sample proportion is 0.55, which of the following could be a 90% confidence interval for the population proportion? Select all that apply.

a. Lower bound: 0.50; Upper bound: 0.60

b. Lower bound: 0.53; Upper bound: 0.59

c. Lower bound: 0.52; Upper bound: 0.58

d. Lower bound: 0.60; Upper bound: 0.70

e. Lower bound: 0.45; Upper bound: 0.60

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.

In a survey of 1000 U.S. adults, 71% think teaching is one of the most important jobs in our country today. The survey’s margin of error is ±3%. (Source: Rasmussen Reports)

The 116th House of Representatives of the United States of America has 435 members, of which 106 are women. An alien lands near the U.S. Capitol and treats members of the House as a random sample of the human race. He reports to his superiors that a 95% confidence interval for the proportion of the human race that is female has a lower bound of 0.203 and an upper bound of 0.284. What is wrong with the alien’s approach to estimating the proportion of the human race that is female? 

In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)

a. Find the point estimate for the population proportion.

Over the first $20$ days of the semester, one student is late to class on $6$ days. Construct a $98\%$ confidence interval for the true proportion of time this student is late. 

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

Why Check It? Why is it necessary to check that np^ ≥ 5 and nq^ ≥ 5?