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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 22. (page 565)

Philly fanatics? Nationally, the proportion of red cars on the road is $0.12 .$ A statistically minded fan of the Philadelphia Phillies (whose team color is red) wonders if Phillies fans are more likely to drive red cars. One day during a home game, he takes a random sample of $210$ cars parked at Citizens Bank Park (the Phillies home field), and counts $35$ red cars.
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 (a) is 0.0187. Interpret the P-value.
d. What conclusion would you make at the α=0.05 significance level?

Short Answer

Expert verified

a. $H_{0} : p = 0.12$ and $H_{1} : p > 0.12$

b. Sample proportion is greater than $0.12$ and it corresponds to claim of alternative hypothesis.

c. There is a $1.87 %$ confidence that $35$ or more out of $210$ randomly selected Phillies fans own a red car.

d. Enough convincing proof is present that the proportion of red cars among Phillies fans is greater than $0.12$ ..

Step by step solution

01

Part (a) Step 1: Given Information

It is given that claim is greater than $0.12$ .

02

Part (a) Step 2: Explanation

am

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Most popular questions from this chapter

Candy! A machine is supposed to fill bags with an average of 19.2 ounces of candy. The manager of the candy factory wants to be sure that the machine does not consistently underfill or overfill the bags. So the manager plans to conduct a significance test at the $α = 0.10$ significance level of

$H_{0} : μ = 19.2 H_{a} : μ n o t e q u a l t o 19.2$

where $μ =$ the true mean amount of candy (in ounces) that the machine put in all bags filled that day. The manager takes a random sample of 75 bags of candy produced that day and weighs each bag. Check if the conditions for performing the test are met.

Error probabilities and power You read that a significance test at the $α = 0.01$

significance level has probability $0.14$ of making a Type II error when a specific alternative is true.

a. What is the power of the test against this alternative?

b. What’s the probability of making a Type I error?

Which of the following is not a condition for performing a significance test about an unknown population proportion p?

(a) The data should come from a random sample or randomized experiment.

(b) Individual measurements should be independent of one another.

(c) The population distribution should be approximately Normal, unless the sample size is large.

(d) Both np and n(1 - p) should be at least 10.

(e) If you are sampling without replacement from a finite population, then you should sample no more than 10% of the population.

$18 %$ Members of the city council want to know if a majority of city residents supports a $1 %$ increase in the sales tax to fund road repairs. To investigate, they survey a random sample of $300$ city residents and use the results to test the following hypotheses:

$H_{0} : p = 0.50$

$H_{a} : p > 0.50$

where $p$ is the proportion of all city residents who support a $1 %$ increase in the sales tax to fund road repairs.

In the sample, $\hat{p} = 158 / 300 = 0.527$ , The resulting $P$ -value is $0.18$ . What is the correct interpretation of this $P$ -value?

a. Only $18 %$ of the city residents support the tax increase.

b. There is an $18 %$ chance that the majority of residents supports the tax increase.

c. Assuming that $50 %$ of residents support the tax increase, there is an $18 %$ probability that the sample proportion would be $0.527$ or greater by chance alone.

d. Assuming that more than $50 %$ of residents support the tax increase, there is an $18 %$ probability that the sample proportion would be $0.527$ or greater by chance alone.

e. Assuming that $50 %$ of residents support the tax increase, there is an $18 %$ chance that the null hypothesis is true by chance alone.

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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