Chapter 10: Comparing Two Populations or Groups

Starting in the $1970s$, medical technology allowed babies with very low birth weight (VLBW, less than $1500$grams, or about $3.3$pounds) to survive without major handicaps. It was noticed that these children nonetheless had difficulties in school and as adults. A long study has followed $242$randomly selected VLBW babies to age $20$years, along with a control group of $233$randomly selected babies from the same population who had normal birth weight.

(a) Is this an experiment or an observational study? Why?

(b) At age $20,179$ of the VLBW group and $193$ of the control group had graduated from high school. Is the graduation rate among the VLBW group significantly lower than for the normal-birth-weight controls? Give appropriate statistical evidence to justify your answer.

To study the long-term effects of preschool programs for poor children, researchers designed an experiment. They recruited $123$children who had never attended preschool from low-income families in Michigan. Researchers randomly assigned $62$of the children to attend preschool (paid for by the study budget) and the other 61 to serve as a control group who would not go to preschool. One response variable of interest was the need for social services as adults. In the past $10$years, $38$children in the preschool group and $49$in the control group have needed social services.

The Minitab output below gives a $95$% confidence interval for $p1-p2$. Interpret this interval in context. Then explain what additional information the confidence interval provides.

A nuclear power plant releases water into a nearby lake every afternoon at $4:51$p.m. Environmental researchers are concerned that fish are being driven away from the area around the plant. They believe that the temperature of the water discharged may be a factor. The scatterplot below shows the temperature of the water (°C) released by the plant and the measured distance (in meters) from the outflow pipe of the plant to the nearest fish found in the water on eight randomly chosen afternoons.

Computer output from a least-squares regression analysis on these data is shown below.

(a) Write the equation of the least-squares regression line. Define any variables you use.

(b) Interpret the slope of the regression line in context.

(c) A residual plot for the regression is shown on the next page. Is a linear model appropriate for describing the relationship between temperature and distance to the nearest fish? Justify your answer.

(d) Does the linear model in part (a) overpredict or underpredict the measured distance from the outflow pipe to the nearest fish found in the water for a temperature of $29°$? Explain your reasoning.

Thirty randomly selected seniors at Council High School were asked to report the age (in years) and mileage of their main vehicles. Here is a scatterplot of the data:

(a) What is the equation of the least-squares regression line? Be sure to define any symbols you use.

(b) Interpret the slope of the least-squares line in the context of this problem. (c) One student reported that her $10$-year-old car had$110000$ miles on it. Find the residual for this data value. Show your work

Drive my car (3.2, 4.3)

(a) Explain what the value of $r^2$tells you about how well the least-squares line fits the data.

(b) The mean age of the students’ cars in the sample was $\overline{x}=8$years. Find the mean mileage of the cars in the sample. Show your work.

(c) Interpret the value of $s$in the context of this setting.

(d) Would it be reasonable to use the least-squares line to predict a car’s mileage from its age for a Council High School teacher? Justify your answer

The Candy Shoppe assembles gift boxes that contain 8 chocolate truffles and two handmade caramel nougats. The truffles have a mean weight of 2 ounces with a standard deviation of 0.5 ounce, and the nougats have a mean weight of 4 ounces with a standard deviation of 1 ounce. The empty boxes weigh 3 ounces with a standard deviation of 0.2 ounce.

(a) Assuming that the weights of the truffles, nougats, and boxes are independent, what are the mean and standard deviation of the weight of a box of candy?

(b) Assuming that the weights of the truffles, nougats, and boxes are approximately Normally distributed, what is the probability that a randomly selected box of candy will weigh more than 30 ounces?

(c) If five gift boxes are randomly selected, what is the probability that at least one of them will weigh more than 30 ounces?

(d) If five gift boxes are randomly selected, what is the probability that the mean weight of the five boxes will be more than 30 ounces?

An investor is comparing two stocks, A and B. She wants to know if over the long run, there is a significant difference in the return on investment as measured by the percent increase or decrease in the price of the stock from its date of purchase. The investor takes a random sample of 50 daily returns over the past five years for each stock. The data are summarized below.

(a) Is there a significant difference in the mean return on investment for the two stocks? Support your answer with appropriate statistical evidence. Use a 5% significance level.

(b) The investor believes that although the return on investment for Stock A usually exceeds that of Stock B, Stock A represents a riskier investment, where the risk is measured by the price volatility of the stock. The standard deviation is a statistical measure of the price volatility and indicates how much an investment’s actual performance during a specified period varies from its average performance over a longer period. Do the price fluctuations in Stock A significantly exceed those of Stock B, as measured by their standard deviations? Identify an appropriate set of hypotheses that the investor is interested in testing.

(c) To measure this, we will construct a test statistic defined $F=\frac{Largersamplevarience}{Smallersamplevarience}$

What value(s) of the statistic would indicate that the price fluctuations in Stock A significantly exceed those of Stock B? Explain.

(d) Calculate the value of the F statistic using the information given in the table.

(e) Two hundred simulated values of this test statistic, F, were calculated assuming no difference in the standard deviations of the returns for the two stocks. The results of the simulation are displayed in the following dot plot.

Use these simulated values and the test statistic that you calculated in part (d) to determine whether the observed data provide evidence that Stock A is a riskier investment than Stock B. Explain your reasoning .

The level of cholesterol in the blood for all men aged $20$to $34$follows a Normal distribution with mean $188$milligrams per deciliter (mg/dl) and a standard deviation $41$mg/dl. For $14$-year-old boys, blood cholesterol levels follow a Normal distribution with a mean $170$mg/dl and a standard deviation of $30$mg/dl.

(a) Let M $=$the cholesterol level of a randomly selected $20$to $34$-year-old man and B =the cholesterol level of a randomly selected $14$-year-old boy. Describe the shape, center, and spread of the distribution of$M-B$

(b) Find the probability that a randomly selected $14$-year-old boy has higher cholesterol than a randomly selected man aged $20$to$34$. Show your work.

The heights of young men follow a Normal distribution with mean of $69.3$inches and standard deviation $2.8$inches. The heights of young women follow a Normal distribution with mean $64.5$inches and standard deviation $2.5$inches.

(a) Let M $=$the height of a randomly selected young man and W $=$the height of a randomly selected young woman. Describe the shape, center, and spread of the distribution of $M-W$

(b) Find the probability that a randomly selected young man is at least $2$inches taller than a randomly selected young woman. Show your work.

Refer exercise $35.$Suppose we select independent SRSs of $25$men aged $20$to $34$and $36$boys aged $14$and calculate the sample mean heights ${\overline{x}}_M$and ${\overline{x}}_B$.

(a) Describe the shape, center, and spread of the sampling distribution of${\overline{x}}_M-{\overline{x}}_B$.

(b) Find the probability of getting a difference in sample means ${\overline{x}}_M-{\overline{x}}_B$that’s less than$0$mg/dl. Show your work.

(c) Should we be surprised if the sample mean cholesterol level for the $14$-year-old boys exceeds the sample mean cholesterol level for the men? Explain.