Chapter 9: Testing a Claim

Stating hypotheses State the appropriate null and alternative hypotheses in each of the following settings. Explain why the sample data give some evidence for $H_a$ in each case.

a. The average height of $18-$year-old American women is $64.2$ inches. You wonder whether the mean height of this year’s female graduates from a large local high school differs from the national average. You measure an SRS of $48$ female graduates and find that $\overline{x}=63.5$ inches and $s_x=3.7$ inches.

b. Rob once read that one-quarter of all people have played/danced in the rain at some point in their lives. His friend Justin thinks that the proportion is higher than $0.25$ for their high school. To settle their dispute, they ask a random sample of $80$ students in their school and find out that $28$ have played/danced in the rain.

Checking conditions Refer to Exercise R9.1. Identify the appropriate test to

perform in each setting and show that the conditions for carrying out the test are met.

Calculations and conclusions Refer to Exercise R9.1. Find the standardized test statistic and $P$-value in each setting, and make an appropriate conclusion.

Fonts and reading ease Does the use of fancy type fonts slow down the reading of text on a computer screen? Adults can read four paragraphs of a certain text in the common Times New Roman font in an average time of 22 seconds. Researchers asked a random sample of 24 adults to read this text in the ornate font named Gigi. Here are their times (in seconds):

Do these data provide convincing evidence that it takes adults longer than 22 seconds, on average, to read these four paragraphs in Gigi font?

Flu vaccine A drug company has developed a new vaccine for preventing the flu. The company claims that fewer than 5% of adults who use its vaccine will get the flu. To test the claim, researchers give the vaccine to a random sample of 1000 adults.

a. State appropriate hypotheses for testing the company’s claim. Be sure to define your parameter.

b. Describe a Type I error and a Type II error in this setting, and give the consequences Page Number: 615 of each.

c. Would you recommend a significance level of 0.01, 0.05, or 0.10 for this test? Justify your choice.

d. The power of the test to detect the fact that only 3% of adults who use this vaccine would develop flu using α=0.05 is 0.9437. Interpret this value.

e. Explain two ways that you could increase the power of the test from part (d).

Flu vaccine Refer to Exercise$R9.5$. Of the $1000$ adults who were given the vaccine, $43$ got the flu. Do these data provide convincing evidence to support the company’s claim?

Roulette An American roulette wheel has $18$red slots among its $38$slots. To test if a particular roulette wheel is fair, you spin the wheel $50$times and the ball lands in a red slot $31$times. The resulting P-value is$0.0384.$

a. Interpret the $P$-value.

b. What conclusion would you make at the$\alpha =0.05$ level?

c. The casino manager uses your data to produce a$99\%$ confidence interval for p and gets$(0.44,0.80)$. He says that this interval provides convincing evidence that the wheel is fair. How do you respond

A milk processor monitors the number of bacteria per milliliter in raw milk received at the factory. A random sample of $10$one-milliliter specimens of milk supplied by one producer gives the following data:

Construct and interpret a 90% confidence interval for the population mean μ.

An opinion poll asks a random sample of adults whether they favor banning ownership of handguns by private citizens. A commentator believes that more than half of all adults favor such a ban. The null and alternative hypotheses you would use to test this claim are

$\text{а.}\mathrm{H}0:{\mathrm{p}}^{\wedge =0.5};\mathrm{Ha}:{\mathrm{p}}^{\wedge >0.5}{H}_0:\hat{p}=0.5;H_a:\hat{p}>0.5\text{.}$

$\text{b. H0:}\mathrm{p}=0.5;\text{Ha:}\mathrm{p}>0.5H_0:p=0.5;H_a:p>0.5\text{.}$

$\text{c. H0:}\mathrm{p}=0.5;\text{Ha:}\mathrm{p}<0.5H_0:p=0.5;H_a:p<0.5\text{.}$

$\text{d. H0:}\mathrm{p}=0.5;\text{Ha:}\mathrm{p}\ne 0.{\mathrm{H}}_0:p=0.5;H_a:p\ne 0.5\text{.}$

$\text{e. H0:}\mathrm{p}>0.5;\text{Ha:}\mathrm{p}=0.5H_0:p>0.5;H_a:p=0.5\text{.}$

Which of the following has the greatest probability?

a.$P(t>2)$if t has 5 degrees of freedom.

b. $P(t>2)$ if t has 2 degrees of freedom.

c. $P(z>2)$ if z is a standard Normal random variable.

d.$P(t<2)$if t has 5 degrees of freedom.

e.$P(z<2)$ if z is a standard Normal random variable.