Conditions for Proportion Confidence Interval

A 95% confidence interval for p, the proportion of all shoppers at a large grocery store who purchase cookies, was found to be (0.236, 0.282).

The point estimate and margin or error for this interval are:

• .236 and .282

• .236 and .046

• unknown and .023

• .259 and .046

• .259 and .023

A 95% confidence interval for p, the proportion of all shoppers at a large grocery store who purchase cookies, was found to be (0.236, 0.282).

Which of the following is a correct statement?

• About 95% of the shoppers have between a 23.6% and a 28.2% chance of purchasing cookies.

• There is a 95% probability that the sample proportion lies between 0.236 and 0.282.

• If a second sample was taken, there is a 95% chance that its confidence interval would contain

  0.25.

• This confidence interval indicates that more than 25% of shoppers buy cookies.

• We are reasonably certain the true proportion of shoppers who purchase cookies is between

  24% and 28%.

A 95% confidence interval for p, the proportion of all shoppers at a large grocery store who purchase cookies, was found to be (0.236, 0.282).

Which of the following would be true about a 98% confidence interval constructed using the same data?

• The interval would be wider, because the standard error would be larger.

• The interval would be narrower, because the standard error would be smaller.

• The interval would be wider, because the critical z* would be larger.

• The interval would be narrower, because the critical z* would smaller.

• The interval would be about the same width, because the standard error would be smaller, but

  the critical z* would be larger.

An SRS of 50 students at a large middle school are asked, “Do you have a television in your bedroom? Twenty-eight of the students responded “yes.”

If ƥ = the proportion of students who answered “yes,” what is the standard error of ƥ ?

• .005

• .010

• .070

• .106

• can't be calculated unless we know standard deviation

The college newspaper of a large Midwestern university periodically conducts a survey of students on campus to determine the attitude on campus concerning issues of interest. Pictures of the students interviewed along with quotes of their responses are printed in the paper. Students are interviewed by a reporter “roaming” the campus selecting students to interview “haphazardly.” On a particular day the reporter interviews five students and asks them if they feel there is adequate student parking on campus. Four of the students say, “no.” Which of the following conditions for inference about a proportion using a confidence interval are violated in this example?

I. The data are an SRS from the population of interest.

II. The population is at least ten times as large as the sample.

III. np̂ ≥ 10 and n(1− p̂) ≥10 .

• I

• II

• II

• I and III

• All 3 conditions are violated

6. You want to estimate the proportion of students who meet the state standards on the SAT with a 90% confidence interval. You don’t have a sample proportion to use yet. If you want the margin of error to be no more than 10%, you will need a minimum sample size of approximately

• 148

• 125

• 68

• 25

• 5

The government claims that 88% of students work during their summer break from college. A random sample of 500 students gave a proportion of 80% and a 95% confidence interval of (.76, .83). Which of the following is a correct interpretation of the 95% level?

• If the study was repeated many times, there is a 95% probability that the true proportion of student that work over the summer is not as the government claims.

• Because 88% is not in our interval, there is a 95% probability that the true proportion of student that work over the summer is not as the government claims.

• If the study was repeated many times, then about 95% of the confidence intervals will contain the true proportion of students that work over the summer college break.

• There is a 95% probability that the true proportion of student that work over the summer is between .76 and 83.

• If the study was repeated many times, then about 95% of the confidence intervals will contain

  .88.

Janice and her cousin Linda are a little competitive about the relative merits of their home towns. One contest they had was to determine who had more rainy days. They found weather records on the internet and each of them randomly selected 60 days from the past 5 years. Janice found that there had been measurable rainfall on 17 of the 60 days she selected for Minooka (M), and Linda found that there had been measurable rainfall on 12 of the 60 days she selected for Green Bay (G). They intend to perform a confidence interval estimating the true difference of proportion of rainy days between Minooka and Green Bay. Which expression should they use to estimate the standard deviation for the sampling distribution of p̂_M - p̂_G

• a

• b

• c

• d

• e

A city planner wants to estimate the proportion of city residents who commute to work by subway each day. A random sample of 30 city residents was selected, and 28 of those selected indicated that they rode the subway to work. Is it appropriate to assume that the sampling distribution of the sample proportion is approximately normal?

• No, because the size of the population is not known.

• No, because the sample is not large enough to satisfy the normality conditions.

• Yes, because the sample is large enough to satisfy the normality conditions.

• Yes, because the sample was selected at random.

• Yes, because sampling distributions of proportions are modeled with a normal model.

Lila and Robert attend different high schools. They will estimate the population percentage of students at their respective schools who have seen a certain movie. Lila and Robert each select a random sample of students from their respective schools and use the data to create a 95 percent confidence interval. Lila’s interval is (0.30,0.35), and Robert’s interval is (0.27,0.34). Which of the following statements can be concluded from the intervals?

• Lila’s sample size is most likely greater than Robert’s sample size

• Robert’s sample size is mostly likely greater than Lila’s sample size

• Lila and Robert will both find the same sample proportion of students who have seen the movie.

• Lila’s interval has a greater degree of confidence than that of Robert.

• Robert’s interval has a greater degree of confidence than that of Lila.