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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset Vaia Explanations$ Math

6th Edition · 837 pages

ISBN: 9781319113339

Chapter 9: Q .6. (page 564)

Ski jump When ski jumpers take off, the distance they fly varies considerably depending on their speed, skill, and wind conditions. Event organizers must position the landing area to allow for differences in the distances that the athletes fly. For a particular competition, the organizers estimate that the variation in distance flown by the athletes will be \(\sigma=10\) \(\sigma=10\) meters. An experienced jumper thinks that the organizers are underestimating the variation.

Short Answer

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01

Step 1:Given information

Ski jump When ski jumpers take off, the distance they fly varies considerably depending on their speed, skill, and wind conditions. Event organizers must position the landing area to allow for differences in the distances that the athletes fly. For a particular competition, the organizers estimate that the variation in distance flown by the athletes will be $\sigma=10$ $\sigma=10$ meters. An experienced jumper thinks that the organizers are underestimating the variation.

state appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

02

Step 2:Explaination

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Most popular questions from this chapter

How much juice? One company's bottles of grapefruit juice are filled by a machine that is set to dispense an average of $180$ milliliters (ml) of liquid. A quality-control inspector must check that the machine is working properly. The inspector takes a random sample of $40$ bottles and measures the volume of liquid in each bottle.

state appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest

Better parking A local high school makes a change that should improve student satisfaction with the parking situation. Before the change, $37 %$ of the school’s students approved of the parking that was provided. After the change, the principal surveys an SRS of students at the school. She would like to perform a test of $H_{0} : p = 0.37 H_{a} : p > 0.37$ where p is the true proportion of students at school who are satisfied with the parking

situation after the change.

a. The power of the test to detect that $p = 0.45$ based on a random sample of $200$ students and a significance level of $α = 0.05$ is $0.75$ Interpret this value.

b. Find the probability of a Type I error and the probability of a Type II error for the test in part (a).

c. Describe two ways to increase the power of the test in part (a).

Clean water The Environmental Protection Agency (EPA) has determined that safe

drinking water should contain at most $1.3$ mg/liter of copper, on average. A water supply company is testing water from a new source and collects water in small bottles at each of $30$ randomly selected locations. The company performs a test at the α=0.05 significance level of $H_{0} : μ = 1.3$ versus $H_{a} : μ > 1.3$ , where μ is the

true mean copper content of the water from the new source.

a. Describe a Type I error and a Type II error in this setting.

b. Which type of error is more serious in this case? Justify your answer.

c. Based on your answer to part (b), do you agree with the company’s choice of $α = 0.05$ ? Why or why not?

Stating hypotheses

a. A change is made that should improve student satisfaction with the parking situation at your school. Before the change, $37 %$ of students approve of the parking that's provided. The null hypothesis $H 0 : p^{\land = 0.37} H_{0} : \hat{p} = 0.37$ is tested against the alternative Ha: $p^{\land > 0.37} H_{a} : \hat{p} > 0.37$

b. A researcher suspects that the mean birth weights of babies whose mothers did not see a doctor before delivery is less than 3000 grams. The researcher states the hypotheses as

$H 0 : μ = 3000 grams H_{0} : μ = 3000 grams$

$Ha: μ \leq 2999 grams H_{a} : μ \leq 2999 grams$

Heavy bread? The mean weight of loaves of bread produced at the bakery where you work is supposed to be $1$ pound. You are the supervisor of quality control at the bakery, and you are concerned that new employees are producing loaves that are too light. Suppose you weigh an SRS of bread loaves and find that the mean weight is $0.975$ pound.

a. State appropriate hypotheses for performing a significance test. Be sure to define the parameter of interest.

b. Explain why there is some evidence for the alternative hypothesis.

c. The P-value for the test in part (a) is $0.0806$ . Interpret the P-value.

d. What conclusion would you make at the $α = 0.01$ significance level?

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