Chapter 10: Comparing Two Populations or Groups

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A study of road rage asked separate random samples of $596$men and $523$women about their behavior while driving. Based on their answers, each re-spondent was assigned a road rage score on a scale of $0$to $20$. Are the conditions for performing a two-sample t test satisfied?

a) Maybe; we have independent random samples, but we need to look at the data to check Normality.

(b) No; road rage scores in a range between $0$ and $20$ can’t be Normal.

(c) No; we don’t know the population standard deviations.

(d) Yes; the large sample sizes guarantee that the corresponding population distributions will be Normal.

(e) Yes; we have two independent random samples and large sample sizes.

Toyota or Nissan? Are Toyota or Nissan owners more satisfied with their vehicles? Let’s design a study to find out. We’ll select a random sample of $400$Toyota owners and a separate random sample of $400$Nissan owners. Then we’ll ask each individual in the sample: “Would you say that you are generally satisfied with your (Toyota/Nissan) vehicle?”

(a) Is this a problem with comparing means or comparing proportions? Explain.

(b) What type of study design is being used to produce data?

Suppose the probability that a softball player gets a hit in any single at-bat is .$300$. Assuming that her chance of getting a hit at a particular time at bat is independent of her other times at bat, what is the probability that she will not get a hit until her fourth time at bat in a game?

(a) $\left(\begin{array}{l}4\\ 3\end{array}\right)(0.3{)}^1(0.7{)}^3$

(b) $\left(\begin{array}{l}4\\ 3\end{array}\right)(0.3{)}^3(0.7{)}^1$

(c) $\left(\begin{array}{c}4\\ 1\end{array}\right)(0.3{)}^3(0.7{)}^1$

(d) $(0.3{)}^3(0.7{)}^1$

(e)role="math" localid="1650371346957" $(0.3{)}^1(0.7{)}^3$

Students’ self-concept Here is SAS output for a study of the self-concept of a random sample of seventh-grade students. The variable SC is the score on the Piers-Harris Self-Concept Scale. The analysis was done to see if male and female students differ in mean self-concept score .

Write a few sentences summarizing the comparison of females and males, as if you were preparing a report for publication

Explain why the conditions for using two-sample z procedures to perform inference about $p1-p2$are not met in the settings

Broken crackers We don’t like to find broken crackers when we open the package. How can makers reduce breaking? One idea is to microwave the crackers for $30$seconds right after baking them. Breaks start as hairline cracks called “checking.” Assign $65$newly baked crackers to the microwave and another $65$to a control group that is not microwaved. After one day, none of the microwave group and $16$ of the control group show checking.^$8$

Researchers are interested in evaluating the effect of a natural product on reducing blood pressure. This will be done by comparing the mean reduction in blood pressure of a treatment (natural product) group and a placebo group using a two-sample t-test. The researchers would like to be able to detect whether the natural product reduces blood pressure by at least $7$points more, on average than the placebo. If groups of size $50$are used in the experiment, a two-sample t-test using role="math" localid="1650436089340" $\alpha =0.01$will have a power of $80\%$to detect a $7$-point difference in mean blood pressure reduction. If the researchers want to be able to detect a $5$-point difference instead, then the power of test

(a) would be less than $80\%$.

(b) would be greater than $80\%$.

(c) would still be $80\%$.

(d) could be either less than or greater than $80\%$, depending on whether the natural product is effective.

(e) would vary depending on the standard deviation of the data.

Let $X$represent the score when a fair six-sided die is rolled. For this random variable,${\mu }_X=3.5$ and ${\sigma }_X=1.71$. If the die is rolled $100$times, what is the approximate probability that the total score is at least$375$?

R10.1. Which procedure? For each of the following settings, say which inference procedure from Chapter 8, 9, or 10 you would use. Be specific. For example, you might say, “Two-sample z test for the difference between two proportions.” You do not need to carry out any procedures.
(a) Do people smoke less when cigarettes cost more? A random sample of $500$smokers was selected. The number of cigarettes each person smoked per day was recorded over a one-month period before a $30\%$ cigarette tax was imposed and again for one month after the tax was imposed.
(b) How much greater is the percent of senior citizens who attend a play at least once per year than the percent of people in their twenties who do so?
Random samples of $100$senior citizens and $100$ people in their twenties were surveyed.
(c) You have data on rainwater collected at $16$ locations in the Adirondack Mountains of New York State. One measurement is the acidity of the
water, measured by pH on a scale of $0$to $14$(the pH of distilled water is $7.0$). Estimate the average acidity of rainwater in the Adirondacks.
(d) Consumers Union wants to see which of two brands of calculator is easier to use. They recruit $100$ volunteers and randomly assign them to two
equal-sized groups. The people in one group use Calculator A and those in the other group use Calculator B. Researchers record the time required for each volunteer to carry out the same series of routine calculations (such as figuring discounts and sales tax, totaling a bill) on the assigned calculator.

Who uses instant messaging? Do younger people use online instant messaging (IM) more often than older people? A random sample of IM users found that $73$of the $158$people in the sample aged $18$to $27$said they used IM more often than email. In the $28$to $39$age group, $26$of $143$people used IM more often than email.$9$ Construct and interpret a $90$% confidence interval for the difference between the proportions of IM users in these age groups who use IM more often than email.