Homework 13

Assume that each of the samples below are a random sample from a population that is approximately normally distributed.

For each of the following problems, perform a hypothesis test by completing the following steps:

  1. State the claim in terms of an equality/inequality involving the population mean.
  2. Determine whether the claim corresponds to a null or alternative hypothesis.
  3. State the null and alternative hypotheses.
  4. Determine the type of evidence, involving the sample mean, which would be evidence against the null hypothesis.
  5. Draw the density curve for \(T\) under the null hypothesis.
  6. Determine the observed value of the \(T\)-statistic \(t_{\text{obs}}\) from the sample properties, and indicate where this value falls under the density curve.
  7. Determine whether the observed \(T\)-statistic gives evidence against the null.
  8. Determine the rejection region for the test statistic at the prescribed significance level (aka False Positive Rate) \(\alpha\).
  9. Sketch the rejection region for the test statistic on the density curve for the test statistic.
  10. Test the null hypothesis using the rejection region for the test statistic.
  11. State your conclusion in the language of the original claim. That is, do not just say "Reject the null hypothesis." or "Do not reject the null hypothesis." Relate your conclusion back to the original claim.

In addition to the hypothesis test, compute a two-sided confidence interval for the population mean using a confidence level of 0.95.

Problems

  1. Vision problems in young children often go undetected, delaying proper care. Researchers examined a new computer-based eye exam screening test. In a study sample of 175 young children, screening times ranged from 23 to 357 seconds, with a mean of 84 seconds and a standard deviation of 43 seconds. The screening system will only be put into place if the average screening time is less than 90 seconds. Test the claim that the average screening time for the system in the population of interest is less than 90 seconds at the 0.05 significance level.
  2. Students in a physiology lab collected 50 wood lice from the university grounds. The average length of the wood lice was 10.5 millimeters, and the standard deviation across their lengths was 2 millimeters. Test the claim that the average length of all wood lice on the university grounds i equal to 10 millimeters at the 0.02 significance level.
  3. The alcohol content of wine depends on the grape variety, the way in which the wine is produced, and other influences. A vineyard in Italy wants to produce wine that has an alcohol content of at most 13 percent on average. A random sample of 48 wines from the vineyard has an average alcohol content of 12.1 percent, and the standard deviation across the wines in the sample is 1.5 percent. Test the claim that the average alcohol content of all wine produced by the vineyard is at most 13 percent at the 0.001 significance level.
  4. Spider silk is the strongest known material, by-weight, currently known. A study examined the mechanical properties of spider silk using a random sample of 21 female golden orb weavers. The average silk yield stress of their spider silk was 205 megapascals, and the standard deviation across their silk yield stress was 50 megapascals. Test the claim that the average silk stress yield of spider silk is 200 megapascals at the 0.1 significance level.