Exam 10: Asking and Answering Questions About a Population Proportion

The larger the sample size, the greater the power.

Let p denote the proportion of houses that are for rent in a neighborhood. For a large-sample z-test of Hp₀ : = 0.35 versus Hp_a : < 0.35 , find the P-value associated witheach of the following values of the z test statistic. a) -0.55 b) -2.80

The two possible conclusions in a hypothesis test are to reject the nullhypothesis or to accept the null hypothesis.

A Type II error is the error of rejecting a true null hypothesis.

Hermit crabs fight for ownership of shells. Fights are initiated when one crab raps onthe shell of another with its big claw. An exchange of shells occurs approximately50% of the time. Investigators wish to see if the rapping force helps decide the issue.They set up an experiment with a rubberized shell, which would dampen the force ofthe rapping. They reason that the reduced force should result in fewer exchanges ofshells. They then observed 59 fights to see how many shell exchanges occurred. a) What null and alternative hypotheses should the investigators use? In a fewsentences, justify your choice of the alternative hypothesis b) Describe a Type I error and a Type II error in this context.

Describe in a few sentences how each of the following affects the power of ahypothesis test. a) The size of the difference between the actual value and the hypothesized value ofthe population proportion. b) The significance level,? c) The sample size

If p = 0.90, a sample size of n =10 is large enough for the large-sampletest for a population proportion to be appropriate.

A hypothesis test is only capable of demonstrating strong support for thealternative hypothesis.

Assuming a random sample from a large population, for which of the following nullhypotheses and sample sizes is the large-sample z test appropriate? Show thecalculations leading to your responses. a) b)

a) 0.35 b) 1.95

The P-value is used to calculate the test statistic.

The smaller the significance level, ?, the greater the power.

The larger the difference between the hypothesized value and the actualvalue of p, the greater the power.

Describe in a few sentences how each of the following affects the power of ahypothesis test. a) The size of the difference between the actual value and the hypothesized value ofthe population proportion. b) The significance level, ? c) The sample size

The probability of a Type I error is denoted by?.

The field of pediatrics has consistently attracted a large number of women. Tobalance their personal and professional lives, many women choose part time pediatricpositions. In the year 2000, it was estimated that 45% of female pediatricians workedpart time. Researchers would like to collect data in order to see if there is evidencethat this percentage has changed in the intervening years. a) What null and alternate hypotheses should the investigators use? In a fewsentences, justify your choice of the alternative hypothesis b) Describe a Type I error and a Type II error in this context.

$\hat{p}$ is a statistic.

Suppose that the study in problem #2 was performed with a random sample ofn =121patient contacts where a plain ring was worn, and 40 of these patient contactsresulted in bacterial transmission. a) Describe the shape, center, and spread of the sampling distribution of $\hat{p}$ if the nullhypothesis Hpo : = 0.25 were true. b) Is there convincing evidence that the null hypothesis is not true, or is $\hat{p}$ consistentwith what you would expect to see if the null hypothesis were true? Carry out ahypothesis test to answer this question.

Bats feed on insects at night and sleep during the day. Many species of bats use theundersides of bridges for sleeping places. The beams that support a bridge create twokinds of spaces in equal numbers: wide (approximately 55 cm) and narrow(approximately 17 cm.) Biologists believe the bats choose narrow spaces to providemore safety from predators. Investigators studying the sleeping position choices ofthe Big-eared Bat (Corynorhinus rafinesquii) observed that 67 out of 102 of themchose narrow beam spaces in which to sleep.Does this sample provide sufficient evidence to conclude that the bats prefer narrowover wide sleeping space? Use a significance level of .05 to test the appropriatehypotheses.

Briefly address the following three questions about testing hypotheses. a) Explain in your own words what a hypothesis test is. b) Explain in your own words the distinction between a null hypothesis and analternative hypothesis c) What are the two possible conclusions when testing a hypothesis?