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The Practice of Statistics for AP Examination

Daren Starnes, Josh Tabor

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 837 pages

ISBN: 9781319113339

Chapter 10: Q 17. (page 641)

Shrubs and fire Fire is a serious threat to shrubs in dry climates. Some shrubs can
resprout from their roots after their tops are destroyed. Researchers wondered if fire would help with resprouting. One study of resprouting took place in a dry area of Mexico. The researchers randomly assigned shrubs to treatment and control groups. They clipped the tops of all the shrubs. They then applied a propane torch to the stumps of the treatment group to simulate a fire. All $12$ of the shrubs in the treatment group resprouted. Only $8$ of the $12$ shrubs in the control group resprouted.
a. State appropriate hypotheses for performing a significance test. Be sure to define the parameters of interest.
b. Check if the conditions for performing the test are met.

Short Answer

Expert verified

a. The hypothesis are $H_{0} : p_{1} = p_{2}$ and $H_{a} : p_{1} > p_{2}$

b. All conditions are not met.

Step by step solution

01

Given Information

It is given that the researchers want to know that if there is a difference in the proportion of shrubs that received the treatment and resprouted and the shrubs that did not received the treatment and resprouted.

$x_{1} = 12$

$x_{2} = 8$

$n_{1} = 12$

$n_{2} = 12$

02

Appropriate hypothesis

Claim is that proportion is greater than shrubs in treatment group.

The appropriate hypothesis is:

Null: $H_{0} : p_{1} = p_{2}$

Alternative: $H_{a} : p_{1} > p_{2}$

$p_{1}$ is proportion of shrubs that received the treatment and resprouted.

$p_{2}$ is proportion of shrubs that did not received the treatment and resprouted.

03

Conditions

Three conditions are:

Random: Independent random samples assigned to shrubs.

Independent: $12$ shrubs is less than $10 %$ of all shrubs.

Normal: There are $12$ success and zero failures, zero is less than ten.

All conditions are not satisfied. We cannot use hypothesis for testing a claim.

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Most popular questions from this chapter

Friday the 13th Do people behave differently on Friday the $13 t h$ ? Researchers collected data on the number of shoppers at a random sample of $45$ grocery stores on Friday the $6 t h$ and Friday the $13 t h$ in the same month. Then they calculated the difference (subtracting in the order $6 t h m i n u s 13 t h$ ) in the number of shoppers at each store on these $2$ days. The mean difference is $- 46.5$ and the standard deviation of the differences is $178.0$ .

a. If the result of this study is statistically significant, can you conclude that the difference in shopping behavior is due to the effect of Friday the $13 t h$ on people鈥檚 behavior? Why or why not?

b. Do these data provide convincing evidence at the $伪 = 0.05$ level that the number of shoppers at grocery stores on these $2$ days differs, on average?

c. Based on your conclusion in part (a), which type of error鈥攁 Type I error or a Type II error鈥攃ould you have made? Explain your answer.

A researcher wished to compare the average amount of time spent in extracurricular activities by high school students in a suburban school district with that in a school district of a large city. The researcher obtained an SRS of 60 high school students in a large suburban school district and found the mean time spent in extracurricular activities per week to be 6 hours with a standard deviation of 3 hours. The researcher also obtained an independent SRS of 40 high school students in a large city school district and found the mean time spent in extracurricular activities per week to be 5 hours with a standard deviation of 2 hours. Suppose that the researcher decides to carry out a significance test of $H_{0} : 渭_{s u b u r b a n} = 渭_{c i t y}$ versus a two-sided alternative. Which is the correct standardized test statistic ?

(a) $z = \frac{(6 - 5) - 0}{\sqrt{\frac{3}{60} + \frac{2}{40}}}$

(b) $z = \frac{(6 - 5) - 0}{\sqrt{\frac{3^{2}}{60} + \frac{2^{2}}{40}}}$

(c) role="math" localid="1654192807425" $t = \frac{(6 - 5) - 0}{\frac{3}{\sqrt{60}} + \frac{2}{\sqrt{40}}}$

(d) $t = \frac{(6 - 5) - 0}{\sqrt{\frac{3}{60} + \frac{2}{40}}}$

(e) $t = \frac{(6 - 5) - 0}{\sqrt{\frac{3^{2}}{60} + \frac{2^{2}}{40}}}$

Drive-thru or go inside? Many people think it鈥檚 faster to order at the drive-thru than to order inside at fast-food restaurants. To find out, Patrick and William used a random number generator to select $10$ times over a $2 鈥�$ week period to visit a local Dunkin鈥� Donuts restaurant. At each of these times, one boy ordered an iced coffee at the drive-thru and the other ordered an iced coffee at the counter inside. A coin flip determined who went inside and who went to the drive-thru. The table shows the times, in seconds, that it took for each boy to receive his iced coffee after he placed the order.

Do these data provide convincing evidence at the $伪 = 0.05$ level of a difference in the true mean service time inside and at the drive-thru for this Dunkin鈥� Donuts restaurant?

Are TV commercials louder than their surrounding programs? To find out, researchers collected data on $50$ randomly selected commercials in a given week. With the television鈥檚 volume at a fixed setting, they measured the maximum loudness of each commercial and the maximum loudness in the first $30$ seconds of regular programming that followed. Assuming conditions for inference are met, the most appropriate method for answering the question of interest is

a. a two-sample t test for a difference in means.

b. a two-sample t interval for a difference in means.

c. a paired t test for a mean difference.

d. a paired t interval for a mean difference.

e. a two-sample z test for a difference in proportions.

Response bias Does the appearance of the interviewer influence how people respond to a survey question? Ken (white, with blond hair) and Hassan (darker, with Middle Eastern features) conducted an experiment to address this question. They took turns (in a random order) walking up to people on the main street of a small town, identifying themselves as students from a local high school, and asking them, 鈥淒o you support President Obama鈥檚 decision to launch airstrikes in Iraq?鈥� Of the $50$ people Hassan spoke to, $11$ said 鈥淵es,鈥� while $21$ of the $44$ people Ken spoke to said 鈥淵es.鈥� Construct and interpret a $90 %$ confidence interval for the difference in the proportion of people like these who would say they support President Obama鈥檚 decision when asked by Hassan versus when asked by Ken.

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