91Ó°ÊÓ

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.

Teens and sex The Gallup Youth Survey asked a random sample of U.S. teens aged 13 to 17 whether they thought that young people should wait until marriage to have sex.14 The Minitab output shows the results of a significance test and a 95% confidence interval based on the survey data.

a. Define the parameter of interest.

b. Check that the conditions for performing the significance test are met in this case.

c. Interpret the P-value.

d. Do these data give convincing evidence that the actual population proportion differs from $0.5$? Justify your answer with appropriate evidence.

After once again losing a football game to the archrival, a college’s alumni association conducted a survey to see if alumni were in favor of firing the coach. An SRS of 100 alumni from the population of all living alumni was taken, and 64 of the alumni in the sample were in favor of firing the coach. Suppose you wish to see if a majority of all living alumni is in favor of firing the coach. The appropriate standardized test statistic is

$$\begin{gathered} \left(a\right)z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{100}}}} \\ z=0.64-0.50.64(0.36)100 \end{gathered}$$

role="math" localid="1654432946823" $$\begin{gathered} \left(b\right)t=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{100}}}} \\ t=0.64-0.50.64(0.36)100 \end{gathered}$$

$$\begin{gathered} \left(c\right)z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}} \\ z=0.64-0.50.5(0.5)100 \end{gathered}$$

$$\begin{gathered} \left(d\right)z=\frac{0.64-0.5}{\sqrt{{\displaystyle \frac{0.64(0.36)}{64}}}} \\ z=0.64-0.50.64(0.36)64 \end{gathered}$$

$$\begin{gathered} \left(e\right)z=\frac{0.5-0.64}{\sqrt{{\displaystyle \frac{0.5(0.5)}{100}}}} \\ z=0.5-0.640.5(0.5)100 \end{gathered}$$

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.

Which of choices (a) through (d) is not a condition for performing a significance test about a population proportion p?

a. The data should come from a random sample from the population of interest.

b. Both $np0$and $n(1-p0)$should be at least $10.$

c. If you are sampling without replacement from a finite population, then you should sample less than $10\%$of the population.

d. The population distribution should be approximately Normal unless the sample size is large.

e. All of the above are conditions for performing a significance test about a population proportion.

The standardized test statistic for a test of $H_0:p=0.4$versus $H_a:pnotequalto0.4$is$z=2.43$This test is

a. not significant at either $\mathrm{Î\pm }=0.05$or $\mathrm{Î\pm }=0.01$

b. significant at $\mathrm{Î\pm }=0.05$but not at$\mathrm{Î\pm }=0.01$

c. significant at$\mathrm{Î\pm }=0.01$but not at $\mathrm{Î\pm }=0.05$

d. significant at both $\mathrm{Î\pm }=0.05$and$\mathrm{Î\pm }=0.01$

e. inconclusive because we don’t know the value of $\hat{p}$

Which of the following $95\%$confidence intervals would lead us to reject $H_0:p=0.30$in favor of $H_a:pnotequalto0.30$at the $5\%$significance level?

a. $(0.19,0.27)$

b.$(0.24,0.30)$

c. $(0.27,0.31)$

d. $(0.29,0.31)$

e. None of these

Packaging DVDs $(6.2,5.3)$ A manufacturer of digital video discs (DVDs) wants to be sure that the DVDs will fit inside the plastic cases used as packaging. Both the cases and the DVDs are circular. According to the supplier, the diameters of the plastic cases vary Normally with mean $\mathrm{μ}=5.3$inches and standard deviation $\mathrm{σ}=0.01$inch. The DVD manufacturer produces DVDs with mean diameter$\mathrm{μ}=5.26$inches. Their diameters follow a Normal distribution with $\mathrm{σ}=0.02$inch.

a. Let X = the diameter of a randomly selected case and Y = the diameter of a randomly selected DVD. Describe the shape, center, and variability of the distribution of the random variable X−Y. What is the importance of this random variable to the DVD manufacturer?

b. Calculate the probability that a randomly selected DVD will fit inside a randomly selected case.

c. The production process runs in batches of 100 DVDs. If each of these DVDs is paired with a randomly chosen plastic case, find the probability that all the DVDs fit in their cases.

Cash to find work? (4.2) Will cash bonuses speed the return to work of unemployed people? The Illinois Department of Employment Security designed an experiment to find out. The subjects were 10,065 people aged 20 to 54 who were filing claims for unemployment insurance. Some were offered \(500 if they found a job within 11 weeks and held it for at least 4 months. Others could tell potential employers that the state would pay the employer \)500 for hiring them. A control group got neither kind of bonus.15

a. Describe a completely randomized design for this experiment.

b. Explain how you would use a random number generator to assign the treatments.

c. What is the purpose of the control group in this setting?

Attitudes The Survey of Study Habits and Attitudes (SSHA) is a

psychological test with scores that range from $0$to$200.$. The mean score for U.S. college students is$115$. A teacher suspects that older students have better attitudes toward school. She gives the SSHA to an SRS of $45$students from the more than $1000$students at her college who are at least $30$years of age. The teacher wants to perform a test at the $\mathrm{Î\pm }=0.05$significance level of

$$\begin{gathered} H_0:\mathrm{μ}=115 \\ {H}_a:\mathrm{μ}>115 \end{gathered}$$

where $\mathrm{μ}=$the mean SSHA score in the population of students at her college who are at least $30$years old. Check if the conditions for performing the test are met.