Exam 10: Asking and Answering Questions About a Population Proportion

The two possible conclusions in a hypothesis test are to reject the null $\underline { hypothesis }$ or $\underline{ to accept }$ the null hypothesis.

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 an alternative hypothesis. c) What are the two possible conclusions when testing a hypothesis?

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

Let $p$ denote the proportion of corn stalks infected by Gibberella Stalk rot. For a large-sample $z$ -test of $H _ { 0 } : p = 0.10$ versus $H _ { a } : p > 0.10$ , find the $\mathrm { P }$ -value associated with each of the following values of the $z$ test statistic. a) $0.35$ b) $1.95$

Let $p$ denote the proportion of houses that are for rent in a neighborhood. For a largesample $z$ -test of $H _ { 0 } : p = 0.45$ versus $H _ { a } : p < 0.45$ , find the P-value associated with each of the following values of the $z$ test statistic. a) -0.65 b) -1.95

Children develop "representational insight," a connection between an object and a symbol for that object. A random sample of 2-year olds were shown a video of someone putting a toy under 1 of 4 randomly placed boxes in a room familiar to the child. Then they were taken to the room, and asked to "find the toy." The investigators reasoned that a child with representational insight would pick the correct box on the first try. If not, they would find the toy on the first try only $25 \%$ of the time. Thirty out of 57 children found the toy by turning over the correct box on the first try. Do these results provide convincing evidence that the proportion of 2-year old children who choose the correct box on the first try is greater than $0.25$ ? Use a significance level of $\alpha = .05$ to test the appropriate hypotheses.

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

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

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, $\alpha$ c) The sample size

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

A statistic is a characteristic of a population.

A hypothesis test uses population data to choose between two competing hypotheses.

A 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) $H _ { 0 } : p = 0.08 , n = 50$ b) $H _ { 0 } : p = 0.20 , n = 200$

Children as young as 2 years of age, upon seeing an object placed under a pillow in a familiar 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 the proportion of 2-year-olds who remember in the laboratory situation. Investigators wish to determine whether the proportion of 2-year-olds who remember where to look is smaller than the know proportion for the home setting when children are in a laboratory 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.