Exam 10: Asking and Answering Questions About a Population Proportion

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample $z$ test appropriate? Show the calculations leading to your responses. a) $H _ { 0 } : p = 0.72 , n = 25$ b) $H _ { 0 } : p = 0.40 , n = 200$

Suppose that the study in problem #2 was performed with a random sample of $n = 200$ children, and 50 of the children remembered where to look for the object in the setting that was away from home. a) Describe the shape, center, and spread of the sampling distribution of $\hat { p }$ if the null hypothesis $H _ { o } : p = 0.35$ were true. b) Is there convincing evidence that the null hypothesis is not true, or is $\hat { p }$ consistent with what you would expect to see if the null hypothesis were true? Carry out a hypothesis test to answer this question.

After assessing the consequences of Type I and Type II errors, you should identify the largest acceptable $\alpha$ .

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

Let $p$ denote the proportion of houses that are for rent in a neighborhood. For a largesample $z$ -test of $H _ { 0 } : p = 0.15$ versus $H _ { a } : p < 0.15$ , find the P-value associated with each of the following values of the $z$ test statistic. a) $- 1.75$ b) $- 0.45$

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

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

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample $z$ test appropriate? Show the calculations leading to your responses. a) $H _ { 0 } : p = 0.16 , n = 50$ b) $H _ { 0 } : p = 0.20 , n = 180$

Hermit crabs fight for ownership of shells. Fights are initiated when one crab raps on the shell of another with its big claw. An exchange of shells occurs approximately 50% 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 of the rapping. They reason that the reduced force should result in fewer exchanges of shells. They then observed 59 fights to see how many shell exchanges occurred. a) What null and alternative hypotheses should the investigators use? In a few sentences, justify your choice of the alternative hypothesis b) Describe a Type I error and a Type II error in this context.

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?

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

Assuming a random sample from a large population, for which of the following null hypotheses and sample sizes is the large-sample z test appropriate? Show the calculations leading to your responses. a) $H _ { 0 } : p = 0.36 , n = 25$ b) $H _ { 0 } : p = 0.10 , n = 200$

There is disagreement among health care professionals about whether health care workers should wear finger rings while performing patient-related work. In particular, plain rings are presumed to have little impact on bacterial transmission by hand. Previous studies have shown that bacteria are transmitted by patient contact in about $25 \%$ of patient contacts where no ring is worn. Let $p$ denote the proportion of bacterial transmissions when a plain ring is worn. Investigators wish to determine whether the proportion of bacterial transmissions when wearing plain rings is greater than $0.25$ . a) What is the appropriate null hypothesis in this study? b) What is the appropriate alternative hypothesis in this study? c) In the context of this study, describe a Type I error and a Type II error.

In an analysis of hunting by African lions, biologists filmed prey captures from the safety of their vehicles. The capture of prey was divided into a sequence of events for study, one of which is the stalk, defined as the reduction of predator-prey distance for prey that has been specifically located and the prey is unaware of or minimally alarmed by the predator. The investigators identified two types of stalk: (a) "crouching," -- the lion is concealed and either the lion advances toward the prey or the prey advances (unaware) toward the lion, and (b) "running," -- the lion is less concealed and advances toward the prey in a rapid manner. Data on lions' stalks of Thomson's and Grant's gazelles from a random sample of 151 kills is summarized in the table below. Characteristic Numeric value Mean stalking time 24.9 Standard deviation of stalk time 3.0 Proportion of stalks of the crouching type 0.79 Researchers believe that the proportion of stalks that are the crouching type is about 0.87. Do the data above provide evidence that the researchers' belief is incorrect and that the proportion of crouching stalks of Thomson's and Grant's gazelles is different from 0.87?

Let $p$ denote the proportion of houses that are for rent in a neighborhood. For a largesample $z$ -test of $H _ { 0 } : p = 0.35$ versus $H _ { a } : p < 0.35$ , find the $\mathrm { P }$ -value associated with each of the following values of the $z$ test statistic. a) $- 0.55$ b) $- 2.80$

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

In an analysis of hunting by African lions, biologists filmed prey captures from the safety of their vehicles. Prey captures were then divided into a sequence of events. One of the events is the stalk, defined as the reduction of predator-prey distance for prey that has been specifically targeted. The investigators identified two types of stalk: (a) "crouching," -- the lion is concealed and either the lion advances toward the prey or the prey advances (unaware) toward the lion, and (b) "running," -- the lion is less concealed and advances toward the prey in a rapid manner. Data on lions' stalks of wildebeests and zebras from a simple random sample of 159 kills are summarized in the table below. Characteristic Numeric value Mean stalking time 31.6 Standard deviation of stalk time 16.4 Proportion of stalks of the crouching type 0.92 Researchers believe that the proportion of stalks that are the crouching type is about 0.87. Do the data above provide evidence that the researchers' belief is incorrect and that the proportion of crouching stalks of wildebeests and zebras is different from 0.87?