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Coaching and SAT scores $(10.2)$ What we really want to know is whether coached students improve more than uncoached students and whether any advantage is large enough to be worth paying for. Use the information above to answer these questions:

(a) Is there good evidence that coached students gained more on average than uncoached students? Carry out a significance test to answer this question.

(b) How much more do coach students gain on the average? Construct and interpret a $99\%$ confidence interval.

(c) Based on your work, what is your opinion: do you think coaching courses are worth paying for?

Quality control $(2.2,5.3,6.3)$Many manufacturing companies use statistical techniques to ensure that the products they make meet standards. One common way to do this is to take a random sample of products at regular intervals throughout the production shift. Assuming that the process is working properly, the mean measurements from these random samples will vary Normally around the target mean $渭$, with a standard deviation of $蟽$. For each question that follows, assume that the process is working properly.

(a) What's the probability that at least one of the next two sample means will fall more than $2蟽$from the target mean $渭$? Show your work.

(b) What's the probability that the first sample mean that is greater than $渭+2蟽$is the one from the fourth sample taken?

(c) Plant managers are trying to develop a criterion for determining when the process is not working properly. One idea they have is to look at the $5$ most recent sample means. If at least $4$of the $5$ fall outside the interval$(渭-蟽,渭+蟽)$, they will conclude that the process isn't working. Is this a reasonable criterion? Justify your answer with an appropriate probability.

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

In-line skatersA study of injuries to in-line skaters used data from the National Electronic Injury Surveillance System, which collects data from a random sample of hospital emergency rooms. The researchers interviewed $161$ people who came to emergency rooms with injuries from in-line skating. Wrist injuries (mostly fractures) were the most common.$6$The interviews found that $53$people were wearing wrist guards and $6$of these had wrist injuries. Of the $108$who did not wear wrist guards, $45$had wrist injuries.

Golf club repairs The Ping Company makes custom-built golf clubs and competes in the $\$4$billion golf equipment industry. To improve its business process, Ping decided to study the time it took to repair golf clubs sent to the company by mail. The company determined that $16\%$of a random sample of orders were sent back to the customers in $5$days or less. Ping examined the processing of repair orders and made changes. Following the changes, $90\%$of a random sample of orders were completed within $5$days. Assume that each of the estimated percent is based on a random sample of n orders.

(a) We used the sample data to construct two $90\%$ confidence intervals for the proportion of orders completed within 5 days-one before and one after the changes at Ping. The two intervals are $(0.858,0.942)$ and $(0.109,0.211)$. Find the value of $n$.

(b) Explain why the interval $(0.858-0.211,0.942-0.109)=(0.647,0.833)$ is not a $95\%$ confidence interval for the improvement in the proportion of orders sent back to customers within $5$ days following the change.

(c) Construct and interpret a correct $95\%$ confidence interval to replace the one in part (b).

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

Shrubs and 铿乺e Fires are a serious threat to shrubs in dry climates. Some shrubs can resprout from their roots after their tops are destroyed. One study of resprouting took place in a dry area of Mexico.$7$The investigators randomly assigned shrubs to treatment and control groups. They clipped the tops of all the shrubs. They then applied a propane torch to the stumps of the treatment group to simulate a 铿乺e. All $12$of the shrubs in the treatment group were resprouted. Only $8$ of the $12$ shrubs in the control group were resprouted.

How much more effective is exercise and drug treatment than drug treatment alone at reducing the rate of heart attacks among men aged $65$and older? To find out, researchers perform a completely randomized experiment involving $1000$healthy males in this age group. Half of the subjects are assigned to receive drug treatment only, while the other half are assigned to exercise regularly and to receive drug treatment. The most appropriate inference method for answering the original research question is

(a) one-sample z test for a proportion.

(b) two-sample z interval for$p_1-p_2$

(c) two-sample z test for $p_1-p_2$.

(d) two-sample t interval for ${渭}_1-{渭}_2$.

(e) two-sample t-test for ${渭}_1-{渭}_2$.

School $A$has $400$students and School $B$has $2700$students. A local newspaper wants to compare the distributions of SAT scores for the two schools. Which of he following would be the most useful for making this comparison?

(a) Back-to-back stemplots for A and B

(b) A scatterplot of A versus B

(c) Dotplots for A and B drawn on the same scale

(d) Two relative frequency histograms of A and B drawn on the same scale

(e) Two frequency histograms for A and B drawn on the same scale