Exam 10: Asking and Answering Questions About a Population Proportion

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) Hp0 : = 0.36, n= 25 b) Hp0 : = 0.10, n= 200

There is disagreement among health care professionals about whether health careworkers should wear finger rings while performing patient-related work. Inparticular, plain rings are presumed to have little impact on bacterial transmission byhand. Previous studies have shown that bacteria are transmitted by patient contact inabout 25% of patient contacts where no ring is worn. Let p denote the proportion ofbacterial transmissions when a plain ring is worn. Investigators wish to determinewhether the proportion of bacterial transmissions when wearing plain rings is greaterthan 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.

Children as young as 2 years of age, upon seeing an object placed under a pillow in afamiliar setting at home, will later remember where to look for about 35% of the time.Investigators believe this capability will be less pronounced in a laboratory situation,where the child is away from the familiar setting of home. Let p denote theproportion of 2-year-olds who remember in the laboratory situation. Investigatorswish to determine whether the proportion of 2-year-olds who remember where to lookis smaller than the know proportion for the home setting when children are in alaboratory situation. 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 thesafety 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 forprey that has been specifically targeted. The investigators identified two types ofstalk: (a) "crouching," -- the lion is concealed and either the lion advances toward theprey or the prey advances (unaware) toward the lion, and (b)"running," -- the lion isless concealed and advances toward the prey in a rapid manner. Data on lions' stalks of wildebeests and zebras from a simple random sample of 159kills are summarized in the table below. 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 andthat the proportion of crouching stalks of wildebeests and zebras is different from 0.87?

A statistic is a characteristic of a population.

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 andanalternative hypothesis c) What are the two possible conclusions when testing a hypothesis?

A hypothesis test uses population data to choose between two competinghypotheses.

Suppose that the study in problem #2 was performed with a random sample ofn = 200 children, and 50 of the children remembered where to look for the object inthe setting that was away from home. a) Describe the shape, center, and spread of the sampling distribution of $\hat{p}$ if the nullhypothesis Hpo : = 0.35 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.

The choice of the alternative hypothesis depends on the objectives of thestudy.

In an analysis of hunting by African lions, biologists filmed prey captures from thesafety of their vehicles. The capture of prey was divided into a sequence of events forstudy, one of which is the stalk, defined as the reduction of predator-prey distance forprey that has been specifically located and the prey is unaware of or minimallyalarmed by the predator. The investigators identified two types of stalk: (a)"crouching," -- the lion is concealed and either the lion advances toward the prey orthe prey advances (unaware) toward the lion, and (b)"running," -- the lion is lessconcealed 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 151kills is summarized in the table below. 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 andthat the proportion of crouching stalks of Thomson's and Grant's gazelles is differentfrom 0.87?

Let p denote the proportion of houses that are for rent in a neighborhood. For a large-sample z-test of Hp₀ : = 0.45 versus Hp_a : < 0.45 , find the P-value associated witheach of the following values of the z test statistic. a) -0.65 b) -1.95

Let p denote the proportion of houses that are for rent in a neighborhood. For a large-sample z-test of Hp₀ : = 0.15 versus Hp_a : < 0.15 , find the P-value associated witheach of the following values of the z test statistic. a) -1.75 b) -0.45

? is also sometimes called the observed significance level.

After assessing the consequences of Type I and Type II errors, you shouldidentify the largest acceptable ?.

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) Hp₀ : = 0.08, n= 50 b) Hp₀ : = 0.20, n= 200

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) Hp₀ : = 0.16, n= 50 b) Hp₀ : = 0.20, n=180