9. Hypothesis Testing for One Sample

"InfidelityAccording to menstuff.org, 22% of married men have “strayed” at least once during their married lives.

a. Describe how you might go about administering a survey to assess the accuracy of this statement.

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"InfidelityAccording to menstuff.org, 22% of married men have “strayed” at least once during their married lives.

b. A survey of 500 married men indicated that 122 have “strayed” at least once during their married life. Construct a 95% confidence interval for the population proportion of married men who have strayed. Use this interval to assess the accuracy of the statement made by menstuff.org.

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[DATA] Filling BottlesA certain brand of apple juice is supposed to have 64 ounces of juice. Because the penalty for underfilling bottles is severe, the target mean amount of juice is 64.05 ounces. However, the filling machine is not precise, and the exact amount of juice varies from bottle to bottle. The quality-control manager wishes to verify that the mean amount of juice in each bottle is 64.05 ounces so that she can be sure that the machine is not over- or underfilling. She randomly samples 22 bottles of juice, measures the content, and obtains the following data:

A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot does not show any outliers.

b. Explain why a level of significance of α = 0.01 is more reasonable than α = 0.1. [Hint: Consider the consequences of incorrectly rejecting the null hypothesis.]

[DATA] Filling BottlesA certain brand of apple juice is supposed to have 64 ounces of juice. Because the penalty for underfilling bottles is severe, the target mean amount of juice is 64.05 ounces. However, the filling machine is not precise, and the exact amount of juice varies from bottle to bottle. The quality-control manager wishes to verify that the mean amount of juice in each bottle is 64.05 ounces so that she can be sure that the machine is not over- or underfilling. She randomly samples 22 bottles of juice, measures the content, and obtains the following data:

A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot does not show any outliers.

a. Should the assembly line be shut down so that the machine can be recalibrated? Use a 0.01 level of significance.

Calcium in RainwaterCalcium is essential to tree growth. In 1990, the concentration of calcium in precipitation in Chautauqua, New York, was 0.11 milligram per liter (mg/L). A random sample of 10 precipitation dates in 2018 results in the following data:

A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot does not show any outliers. Does the sample evidence suggest that calcium concentrations have changed since 1990? Use the α = 0.05 level of significance.

A nutritionist believes the average daily protein intake for adults is $50g$. A random sample of $37$ adults has a sample mean of $51g$ sample standard deviation of $6g$. Use $\alpha =0.05$ to test whether the true mean protein intake differs from $50$ grams.

A fitness researcher believes a new workout program increases average treadmill endurance beyond $35\min$. A sample of $16$ adults who completed the program had the following endurance times. Test whether the data support the researcher's claim using $\alpha =0.05$ & $\sigma =2.16$.

Designing a Study Stock fund managers are investment professionals who decide which stocks should be part of a portfolio. In an article in the Wall Street Journal (“Not a Stock-Picker’s Market,” WSJ, January 25, 2014), the performance of stock fund managers was considered based on dispersion in the market. In the stock market, risk is measured by the standard deviation rate of return of stock (dispersion). When dispersion is low, then the rate of return of the stocks that make up the market are not as spread out. That is, the return on Company X is close to that of Y is close to that of Z, and so on. When dispersion is high, then the rate of return of stocks is more spread out; meaning some stocks outperform others by a substantial amount. Since 1991, the dispersion of stocks has been about 7.1%. In some years, the dispersion is higher (such as 2001 when dispersion was 10%), and in some years it is lower (such as 2013 when dispersion was 5%). So, in 2001, stock fund managers would argue, one needed to have more investment advice in order to identify the stock market winners, whereas in 2013, since dispersion was low, virtually all stocks ended up with returns near the mean, so investment advice was not as valuable.

e. Suppose this study was conducted and the data yielded a P-value of 0.083. Explain what this result suggests.

Platelet-Rich Plasma In a prospective cohort study, 20 patients with alopecia (hair loss) had platelet-rich plasma (PRP) injected in their scalps. After three months, the mean difference in hair density (after - before) was 170.70 hairs per square centimeter with a standard deviation of 37.81hairs/cm2. Source: Gkini MA, Kouskoukis AE, Tripsianis G, Rigopoulos D, Kouskoukis K., “Study of Platelet-Rich Plasma Injections in the Treatment of Androgenetic Alopecia through a One-Year Period”. J Cutan Aesthet Surg, 2014; 7:213–219. 

c. State the null and alternative hypotheses to determine if hair density increased. 

Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?

Cotinine in Smokers The claim is that smokers have a mean cotinine level greater than the level of 2.84 ng/mL found for nonsmokers. (Cotinine is used as a biomarker for exposure to nicotine.) The sample size is n = 902 and the test statistic is t = 56.319.

Discarded Plastic

What distribution is used for the hypothesis test described in Exercise 1?

For the hypothesis test described in Exercise 1, is it necessary to determine whether the 62 weights appear to be from a population having a normal distribution? Why or why not?