Exam 12: Asking and Answering Questions About a Population Mean

A recent survey sampled 1,086 parents of school-aged children. One question on the survey asked respondents how much time per month (in hours) they spent volunteering at their children's school during the previous school year. The following summary statistics for time volunteered per month were given: ​ ​ Which of the following represents the 98% confidence interval? ​

A boat manufacturer claims that a particular boat and motor combination will burn less than 4.0 gallons of fuel per hour. Fuel consumption for a random sample of 10 similar boats resulted in the data below: 4.06, 4.29, 4.26, 4.64, 4.23, 3.93, 3.64, 4.13, 3.93, 3.86 Is there sufficient evidence to conclude that the manufacturer's claim is correct? Use α = .05 and test the appropriate hypothesis.

Researchers conducted a survey among 1793 families of a certain city who reported that they participated in charity one or more times in 2016. The sample was selected in a way to be representative for families of a certain city who participated in charity in 2016. For this sample, the mean amount of money spent on charity was $876. Suppose that the sample standard deviation was $514. Find the margin of error and construct a 95% confidence interval for the mean amount of money spent in the certain city on charity in 2016.

A random sample of the houses in a particular city is selected and the level of radon gas is measured for each house in the sample. The values collected are given below in parts per million (ppm). Experience has shown that radon gas level is approximately normally distributed for this population. 4, 8, 13, 6, 7, 3, 9, 8, 11, 8 ​ a)Calculate an estimate of the population mean of radon gas levels. b)Construct and interpret a 99% confidence interval for μ. c)The city standards for radon call for no more than a level of 6 ppm. From these data, does it appear that the mean radon level for city houses is under 6 ppm? Provide statistical justification for your response.

The spread of the sample mean decreases when ​

A report states that the mean yearly salary offer for students graduating with accounting degrees in 2012 was $49,007. Suppose that a random sample of 60 accounting graduates at a large university who received job offers resulted in a mean offer of $49,800 and a standard deviation of $3,100. What hypotheses should be used to test whether the accounting graduates at this school are securing starting salaries greater than $49,007? ​

A company provides portable walkie-talkies to construction crews. Their batteries last, on average, 55 hours of continuous use. The purchasing manager receives a brochure advertising a new brand of batteries with a lower price, but suspects that the lifetime of the batteries may be shorter than the brand currently in use. To test this, the new brand is installed in 8 randomly selected radios. Here are the results for the lifetime of the batteries (in hours): 45 52 56 55 51 57 48 52 Is there sufficient evidence to conclude that the purchasing manager is correct in his conjecture that the new brand has a shorter average lifetime?

Under what conditions is the z confidence interval an appropriate way to estimate a population mean?

Researchers studied how many steps a day adult residents of the city make. They determined that the mean number of steps per day for a representative sample of 10 adult residents of the city was 5,139 steps. The original sample data values are: ​ Researchers are interested in deciding if there is evidence that adult residents of the city make more than 5,000 steps per day. Use a randomization test to select the appropriate output for one set of 1,000 simulated sample means and carry out a hypothesis test for a population mean.

A packaging machine is supposed to fill boxes of breakfast cereal to a weight of 16 ounces. Eight filled boxes are selected at random and the weight of cereal is determined. The results (in ounces) are summarized in the following table. A technician wishes to test the claim that the machine is overfilling the boxes. At the 5% significance level, what is the P-value of the test statistic? (Assume the filling weights are normally distributed.)

The Environmental Protection Agency (EPA) is charged with monitoring the environment. One aspect of this is keeping track of "acid rain," a broad term describing the fall of water through an acidic atmosphere. Acidity is measured on the pH scale, where pure water has a pH of 7.0. Normal rain is slightly acidic because carbon dioxide dissolves into it, and thus has a pH of about 5.5. (A lower pH indicates greater acidity.) Suppose the EPA wishes to determine whether a particular area is subject to acid rain. Let μ denote the true average for pH in this area. a)What is the appropriate null hypothesis? b)What is the appropriate alternative hypothesis? c)In your own words, distinguish between a Type I and a Type II error in this context.