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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$\mathrm{Î\pm }=0.05$ 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

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

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

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

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

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

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.

1 A software company is trying to decide whether to produce an upgrade of one of its programs. Customers would have to pay $\backslash (100$ for the upgrade. For the upgrade to be profitable, the company must sell it to more than $20\%$ of their customers. You contact a random sample of $60$ customers and find that $16$ would be willing to pay $\backslash )100$ for the upgrade.

a. Do the sample data give convincing evidence that more than $20\%$ of the company’s customers are willing to purchase the upgrade? Carry out an appropriate test at the $\mathrm{Î\pm }=0.05$significance level.

b. Which would be a more serious mistake in this setting—a Type I error or a Type II error? Justify your answer.

c. Suppose that 30% of the company’s customers would be willing to pay $\$100$ for the upgrade. The power of the test to detect this fact is$0.60.$ Interpret this value.

According to the Bureau of Labor Statistics, the average age of American workers is $41.9$years. The manager of a large technology company believes that the company’s employees tend to be younger, on average. So she takes a random sample of 12 employees and records their ages.

Here are the data:

27 38 32 24 30 47 42 38 27 43 37 33

a. State appropriate hypotheses for testing the manager’s belief. Be sure to define the parameter of interest.

b. State the conditions for performing a test of the hypotheses in (a), and determine whether each condition is met.

c. The P-value of the test is$0.003$. Interpret this value. What conclusion would you make?

A government report says that the average amount of money spent per U.S. household per week on food is about $\backslash (158$. A random sample of$50$ households in a small city is selected, and their weekly spending on food is recorded. The sample data have a mean of \)165 and a standard deviation of $\backslash (20$. Is there convincing evidence that the mean weekly spending on food in this city differs from the national figure of $\backslash )158$?