8.1 Interval Basics
Identify the point estimator you would use to estimate the parameter and calculate the value of the point estimate:
What proportion of students are willing to report cheating by other students? A student project put this question to an SRS of 172 undergraduates at a large university: “You witness two students cheating on a quiz. Do you go to the professor?” Only 19 answered “Yes.”
Check whether the conditions are met for a phat interval:
Latoya wants to estimate the proportion of the seniors at her boarding school who like the cafeteria food. She interviews an SRS of 50 of the 175 seniors and finds that 14 think the cafeteria food is good.
What critical value t* should be used for a confidence interval for the population mean in the following situation?
A 95% confidence interval based on n = 10 randomly selected observations
A Gallup poll asked a random sample of U.S. adults, “Would you like to lose weight?” Based on this poll, the 95% confidence interval for the population proportion who want to lose weight is (0.56, 0.62)
Interpret the interval.
Check whether the conditions have been met for a CI:
A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt
Check if the conditions to make an interval are met:
How much time do students at your school spend on the Internet? You collect data from the 32 members of your AP Statistics class and calculate the mean amount of time that these students spent on the Internet yesterday.
t A Gallup poll asked a random sample of U.S. adults, “Would you like to lose weight?” Based on this poll, the 95% confidence interval for the population proportion who want to lose weight is (0.56, 0.62)
What was the point estimate and margin of error?
When constructing a confidence interval for a population proportion, we check that the sample size is less than 10% of the population size.
a. Why is it necessary to check this condition?
b. What happens to the capture rate if this condition is violated?
A medical study finds that x¯=114.9 and sx = 9.3 for the seated systolic blood pressure of the 27 randomly selected adults. What is the standard error of the mean? Interpret this value in context.
t A Gallup poll asked a random sample of U.S. adults, “Would you like to lose weight?” Based on this poll, the 95% confidence interval for the population proportion who want to lose weight is (0.56, 0.62)
Based on this poll, Gallup claims that more than half of U.S. adults want to lose weight. Use the confidence interval to evaluate this claim.
According to a recent Pew Research Center report, many American adults have made money by selling something online. In a random sample of 4579 American adults, 914 reported that they earned money by selling something online in the previous year. Construct and interpret a 98% CI for the true proportion of American adults that made money by selling something online.
Fruit flies are used frequently in genetic research because of their quick reproductive cycle. The length of the thorax (in millimeters) was measured for each fly in a random sample of 49 male fruit flies. The mean length was x¯=0.8004 mm , with a standard deviation of sx = 0.0782 mm. 29 Construct and interpret a 90% confidence interval for the true mean thorax length of a male fruit fly.
The admissions director for a university found that (107.8, 116.2) is a 95% confidence interval for the mean IQ score of all freshmen. Discuss whether each of the following explanations is correct, based on that information.
This interval was constructed using a method that produces intervals that capture the true mean in 95% of all possible samples.
A college student organization wants to start a nightclub for students under the age of 21. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. What sample size is required to obtain a 90% confidence interval with a margin of error of at most 0.04?
The body mass index (BMI) of all American young women is believed to follow a Normal distribution with a standard deviation of about 7.5. How large a sample would be needed to estimate the mean BMI μ in this population to within ±1 with 99% confidence?