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Marketing The proportion of adult women in the United States is approximately \(51 \%\). A marketing survey telephones 400 people at random. a. What proportion of the sample of 400 would you expect to be women? b. What would the standard deviation of the sampling distribution be? c. How many women, on average, would you expect to find in a sample of that size?

Send money When they send out their fundraising letters, a philanthropic organization typically gets a return from about \(5 \%\) of the people on their mailing list. To see what the response rate might be for future appeals, they did a simulation using samples of size \(20,50,100,\) and 200 . For each sample size, they simulated 1000 mailings with success rate \(p=0.05\) and constructed the histogram of the 1000 sample proportions, shown below. Explain what these histograms show about the sampling distribution model for sample proportions. Be sure to talk about shape, center, and spread.

Campus sample For her final project, Stacy plans on surveying a random sample of 50 students on whether they plan to go to Florida for spring break. From past years, she guesses that about \(10 \%\) of the class goes. Is it reasonablefor her to use a Normal model for the sampling distribution of the sample proportion? Why or why not?

Spanking In a 2015 Pew Research study on trends in marriage and family (www.pewsocialtrends.org/2015/12/17/1the-american-family-today/), \(53 \%\) of randomly selected parents said that they never spank their children. The \(95 \%\) confidence interval is from \(50.6 \%\) to \(55.4 \%(n=1807)\). a. Interpret the interval in this context. b. Explain the meaning of "95\% confident" in this context.

Hiring In preparing a report on the economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days. a. How many randomly selected employers must we contact in order to create an estimate in which we are \(98 \%\) confident with a margin of error of \(5 \% ?\) b. Suppose we want to reduce the margin of error to \(3 \%\). What sample size will suffice? C. Why might it not be worth the effort to try to get an interval with a margin of error of only \(1 \% ?\)

Margin of error A TV newscaster reports the results of a poll of voters, and then says, "The margin of error is plus or minus \(4 \%\)." Explain carefully what that means.

Another margin of error A medical researcher estimates the percentage of children exposed to lead-based paint, adding that he believes his estimate has a margin of error of about \(3 \%\). Explain what the margin of error means.

Conditions For each situation described below, identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a. Police set up an auto checkpoint at which drivers are stopped and their cars inspected for safety problems. They find that 14 of the 134 cars stopped have at least one safety violation. They want to estimate the percentage of all cars that may be unsafe. b. A TV talk show asks viewers to register their opinions on prayer in schools by logging on to a website. Of the 602 people who voted, 488 favored prayer in schools. We want to estimate the level of support among the general public. c. A school is considering requiring students to wear uniforms. The PTA surveys parent opinion by sending a questionnaire home with all 1245 students; 380 surveys are returned, with 228 families in favor of the change. d. A college admits 1632 freshmen one year, and four years later, 1388 of them graduate on time. The college wants to estimate the percentage of all their freshman enrollees who graduate on time.

More conditions Consider each situation described. Identify the population and the sample, explain what \(p\) and \(\hat{p}\) represent, and tell whether the methods of this chapter can be used to create a confidence interval. a.A consumer group hoping to assess customer experiences with auto dealers surveys 167 people who recently bought new cars; \(3 \%\) of them expressed dissatisfaction with the salesperson. b. What percent of college students have cell phones? 2883 students were asked as they entered a football stadium, and 2430 said they had phones with them. c. Two hundred forty potato plants in a field in Maine are randomly checked, and only 7 show signs of blight. How severe is the blight problem for the U.S. potato industry? d. Twelve of the 309 employees of a small company suffered an injury on the job last year. What can the company expect in future years?

Conclusions A catalog sales company promises to deliver orders placed on the Internet within 3 days. Follow-up calls to a few randomly selected customers show that a \(95 \%\) confidence interval for the proportion of all orders that arrive on time is \(88 \% \pm 6 \%\). What does this mean? Are these conclusions correct? Explain. a. Between \(82 \%\) and \(94 \%\) of all orders arrive on time. b. Ninety-five percent of all random samples of customers will show that \(88 \%\) of orders arrive on time. c. Ninety-five percent of all random samples of customers will show that \(82 \%\) to \(94 \%\) of orders arrive on time. d. We are \(95 \%\) sure that between \(82 \%\) and \(94 \%\) of the orders placed by the sampled customers arrived on time. e. On \(95 \%\) of the days, between \(82 \%\) and \(94 \%\) of the orders will arrive on time.