Testing a Claim

(a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise 75? Why?

(b) Explain two ways that the researchers could have increased the power of the test to detect $\mu$=0.5.

The power of tomatoes The researchers who carried out the experiment in Exercise $75$ suspect that the large P-value ($0.114$) is due to low power.

(a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise $75$? Why?

(b) Explain two ways that the researchers could have increased the power of the test to detect .

(a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise 76? Why?

(b) Which of the following changes would give the test a higher power to detect $\mu$=120 minutes: using $\alpha$=0.01 or $\alpha$=0.10? Explain.

Study more! The significance test in Exercise $76$ yields a P-value of $0.0622$.

(a) Describe a Type I and a Type II error in this setting. Which type of error could you have made in Exercise $76$? Why?

(b) Which of the following changes would give the test a higher power to detect $\mu = 120$ minutes: using $\alpha = 0.01$ or $\alpha = 0.10$? Explain.

Pressing pills A drug manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each batch of tablets produced is measured to control the compression process. The target value for the hardness is $\mu$=11.5. The hardness data for a random sample of 20 tablets are

11.627   11.613   11.493   11.602   11.360 

11.374   11.592  11.458   11.552   11.463 

11.383   11.715   11.485   11.509   11.429 

11.477   11.570  11.623   11.472   11.531

Is there significant evidence at the 5% level that the mean hardness of the tablets differs from the target value? Carry out an appropriate test to support your answer.

Pressing pills A drug manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each batch of tablets produced is measured to control the compression process. The target value for the hardness is $\mu = 11.5$. The hardness data for a random sample of $20$ tablets are

$11.627 11.613 11.493 11.602 11.360 11.374 11.592 11.458 11.552 11.463 11.383 11.715 11.485 11.509 11.429 11.477 11.570 11.623 11.472 11.531$

Is there significant evidence at the $5\%$  level that the mean hardness of the tablets differs from the target value? Carry out an appropriate support your answer.

Bottles of a popular cola are supposed to contain $300$ milliliters (ml) of cola. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. From experience, the distribution of the contents is approximately Normal. An inspector measures the contents of six randomly selected bottles from a single day’s production. The results are $299.4 297.7 301.0 298.9 300.2 297.0$ Do these data provide convincing evidence that the mean amount of cola in all the bottles filled that day differs from the target value of $300 ml$? Carry out an appropriate test to support your answer 

Filling cola bottles Bottles of a popular cola are supposed to contain $300$ milliliters (ml) of cola. There is some variation from bottle to bottle because the filling machinery is not perfectly precise. From experience, the distribution of the contents is approximately Normal. An inspector measures the contents of six randomly selected bottles from a single day’s production. The results are $299.4 297.7 301.0 298.9 300.2 297.0$

Refer to Exercise 79. Construct and interpret a $95\%$confidence interval for the population mean M. What additional information does the confidence interval provide .

Refer to Exercise 80. Construct and interpret a $95\%$ confidence interval for the population mean $M$. What additional information does the confidence interval provide? 

How long does it take for a chunk of information to travel from one server to another and back on the Internet? According to the site internettrafficreport.com, a typical response time is $200$ milliseconds (about one-fifth of a second). Researchers collected data on response times of a random sample of$14$servers in Europe. A graph of the data reveals no strong skewness or outliers. The figure below displays Minitab output for a one-sample t interval for the population mean. Is there convincing evidence at the $5\%$ significance level that the site’s claim is incorrect? Use the confidence interval to justify your answer.

A blogger claims that U.S. adults drink an average of five 8-ounce glasses of water per day. Skeptical researchers ask a random sample of 24 U.S. adults about their daily water intake. A graph of the data shows a roughly symmetric shape with no outliers. The figure below displays Minitab output for a one-sample t interval for the population mean. Is there convincing evidence at the $10\%$ significance level that the blogger’s claim is incorrect? Use the confidence interval to justify your answer. 

The P-value for a two-sided test of the null hypothesis $H_0:\mu =10$ is $0.06$ 

(a) Does the$95\%$ confidence interval for$M$ include $10$? Why or why not?

(b) Does the $90\%$ confidence interval for$M$ include $10$? Why or why not

Tests and CIs The P-value for a one-sided test of the null hypothesis$H_0:\mu =15$ is $0.03$.

(a) Does the $99\%$ confidence interval for $\mu$ include $15$ ? Why or why not?

(b) Does the $95\%$ confidence interval for $\mu$include $15$? Why or why not?

Does this paper give convincing evidence that the mean amount of sugar in the hindguts under these conditions is not equal to $7 mg$? Justify your answer. 

Would the hypothesis that $\mu =5\mathrm{mg}$ be rejected at the $5\%$ level in favor of a two-sided alternative? Justify your answer.

The design of controls and instruments affects how easily people can use them. A student project investigated this effect by asking $25$ right-handed students to turn a knob (with their right hands) that moved an indicator. There were two identical instruments, one with a right-hand thread (the knob turns clockwise) and the other with a left-hand thread (the knob must be turned counterclockwise). Each of the $25$ students used both instruments in a random order. The following table gives the times in seconds each subject took to move the indicator a fixed distance:

(a) Explain why it was important to randomly assign the order in which each subject used the two knobs. 

(b) The project designers hoped to show that right-handed people find right-hand threads easier to use. Carry out a significance test at the $5\%$ significance level to investigate this claim 

We hear that listening to Mozart improves students’ performance on tests. Maybe pleasant odors have a similar effect. To test this idea, $21$ subjects worked two different but roughly equivalent paper-and-pencil mazes while wearing a mask. The mask was either unscented or carried a floral scent. Each subject used both masks, in a random order. The table below gives the subjects’ times with both masks.

Does listening to music while studying help or hinder students’ learning? Two AP Statistics students designed an experiment to find out. They selected a random sample of $30$ students from their medium-sized high school to participate. Each subject was given$10$minutes to memorize two different lists of $20$ words, once while listening to music and once in silence. The order of the two word lists was determined at random; so was the order of the treatments. A boxplot of the differences in the number of words recalled (music- silent) is shown below, along with some Minitab output from a one-sample t test. Perform a complete analysis of the students’ data. Include a confidence interval.

Charles Darwin, author of On the Origin of Species (1859), designed an experiment to compare the effects of cross-fertilization and self fertilization on the size of plants. He planted pairs of very similar seedling plants, one self-fertilized and one cross-fertilized, in each of pot$15$ at the same time. After a period of time, Darwin measured the heights (in inches) of all the plants. Here are the data:

(a) Explain why it is not appropriate to perform a paired t test in this setting. 

(b) A hasty student generates the Minitab output shown below. What conclusion should he draw at the $\alpha =0.05$ significance level? Explain 

Is it significant? For students without special preparation, SAT Math scores in recent years have varied Normally with mean $\mu =518$. One hundred students go through a rigorous training program designed to raise their SAT Math scores by improving their mathematics skills. Use your calculator to carry out a test of

$H_0:\mu =518$

$H_{\alpha }:\mu >518$

in each of the following situations.

(a) The students' scores have mean $\overline{x}=536.7$ and standard deviation $s_x=114$. Is this result significant at the$5\%$ level?

(b) 'The students' scores have mean $\overline{x}=537.0$ and standard deviation $s_x=114$. Is this result significant at the $5\%$ level?

(c) When looked at together, what is the intended lesson of (a) and (b)?

Significance and sample size A study with $5000$ subjects reported a result that was statistically significant at the $5\%$ level. Explain why this result might not be particularly large or important.

Sampling shoppers A marketing consultant observes $50$ consecutive shoppers at a supermarket, recording how much each shopper spends in the store. Explain why it would not be wise to use these data to carry out a significance test about the mean amount spent by all shoppers at this supermarket.

Ages of presidents Joe is writing a report on the backgrounds of American presidents. He looks up the ages of all the presidents when they entered office. Because Joe took a statistics course, he uses these numbers to perform a significance test about the mean age of all U.S. presidents. Explain why this makes no sense.

Do you have ESP? A researcher looking for evidence of extrasensory perception (ESP) tests $500$ subjects. Four of these subjects do significantly better $\left(P<0.01\right)$ than random guessing.

(a) Is it proper to conclude that these four people have ESP? Explain your answer.

(b) What should the researcher now do to test whether any of these four subjects have ESP?

What is significance good for? Which of the following questions does a significance test answer? Justify your answer.

(a) Is the sample or experiment properly designed?

(b) Is the observed effect due to chance?

(c) Is the observed effect important?

The reason we use $t$ procedures instead of $z$ procedures when carrying out a test about a population mean is that

(a) $z$ can be used only for large samples.

(b)$z$ requires that you know the population standard deviation $\sigma$.

(c) $z$ requires you to regard your data as an SRS from the population.

(d) $z$ applies only if the population distribution is perfectly Normal.

(e) $z$ can be used only for confidence intervals.

You are testing $H_O:\mu =10$$H_{\alpha }:\mu <10$ against  based on an SRS of $20$ observations from a Normal population. The $t$ statistic is . $t=-2.25$The $P$-value

(a) falls between $0.01 and 0.02$

(b) falls between$0.02 and 0.04$

(c) falls between$0.04 and 0.05$

(d) falls between $.05 and 0.25$.

(c) is greater than$0.25$.

You are testing $H_0:\mu =10$ against $H_{\alpha }:\mu \ne 10$ based on an SRS of $15$$t$ observations from a Normal population. What values of the   statistic are statistically significant at the$\alpha =0.005$ level?

(a) $t>3.326$

(b) $t>3.286$

(c) $t>2.977$

(d) $t<-3.326 or t>3.326$

(c) $t<-3.286 or t>3.286$

After checking that conditions are met, you perform a significance test of  $H_0:\mu =1$versus $H_{\alpha }:\mu \ne 1$. You obtain a $P$value of $0.022$. Which of the following is true?

(a) A$95\%$ confidence interval for $\mu$ will include the value $1$

(b) A $95\%$ confidence interval for$\mu$ will include the value $0$ .

(c)  A $99\%$  confidence interval for $\mu$ will include the value $1$

(d) A $99\%$ confidence interval for $\mu$will include the value $0$

(e) None of these is necessarily true.

 Improving health A large company's medical director launches a health promotion campaign to encourage employees to exercise more and eat better foods. One measure of the effectiveness of such a program is a drop in blood pressure. The director chooses a random sample of $50$ employees and compares their blood pressures from physical cams given before the campaign and again a year later. The mean change (after - before) in systolic blood pressure for these $50$ employees is $-6$and the standard deviation is $19.8$.

(a) Do these data provide convincing evidence of an average decrease in blood pressure among all of the company's employees during this year? Carry out a test at the $\alpha =0.05$ significance level.

(b) Can we conclude that the health campaign caused a decrease in blood pressure? Why or why not?

Does Friday the $13th$ have an effect on people's behavior? Researchers collected data on the numbers of shoppers at a sample of $45$ different grocery stores on Friday the $6th$ and Friday the $13th$ in the same month. The dotplot and computer output below summarize the data on the difference in the number of shoppers at each store on these two days (subtracting in the order $6th$ minus $13th$ ).

Researchers would like to carry out a test of $H_0:{\mu }_d=0$versus $H_{\alpha }:{\mu }_d\ne 0$, where ${\mu }_d$is the true mean difference in the number of grocery shoppers on these two days. Which of the following conditions for performing a paired $t$ test is not met?

1. Random

II. Normal

III. Independent

(a) I only

(b) II only

(c) III only

d) I and II only 

(e) I, II, and III

The most important condition for sound conclusions from statistical inference is that

(a) the data come from a well-designed random sample or randomized experiment

(b) the population distribution be exactly Normal.

(c) the data contain no outliers.

(d) the sample size be no more than $10\%$ of the population size.

(c) the sample size be at least $30$ .

Is your food safe? Do you feel confident or not confident that the food available at most grocery stores is safe to eat? when a gallup poll asked this question, $87\%$of the sample said they were confident." Callup announced the poll's margin of error for $95\%$confidence as$\pm 3$  percentage points. Which of the following sources of error are included in this margin of error? Explain.

(a) Gallup dialed landline telephone numbers at random and so missed all people without landline phones, including people whose only phone is a cell phone.

(b) Some people whose numbers were chosen never answered the phone in several calls or answered but refused to participate in the poll.

(c) There is chance variation in the random selection of telephone numbers.

Spinning for apples (6,3 or 7.3) In the "Ask Marilyn" column of Parade magzine, a reader posed this question: "Say that a slot machine has five wheels, and each wheel has five symbols: an apple, a grape, a peach, a pear, and a plum. I pull the lever five times. What are the chances that I'll get at least one apple?" Suppose that the wheels spin independently and that the fre symbols are equally likely to appear on each wheel in a given spin.

(a) Find the probability that the slot player gets at least one apple in one pull of the lever. Show your method clearly.

(b) Now answer the reader's question. Show your method clearly.

Normal body temperature (8.3) Check that the conditions are met for performing inference about the mean body temperature in the population of interest.

Normal body temperature (8.2) If "normal" body temperature really is $98.6 F$, we would expect the proportion $p$ of all healthy 18- to 40 -year-olds who have body temperatures less than this value to be $0.5$. Construct and interpret a $95\%$ confidence interval for $p$. What conclusion would you draw?

Stating hypotheses State the appropriate null and alternative hypotheses in each of the following cases.

(a) The average height of $18$ -year-old American women is $64.2$ inches. You wonder whether the mean height of this year's female graduates from a large local high school (over $3000$ students) differs from the national average. You measure an SRS of $48$ female graduates and find that $\overline{X}=63.1$ inches.

(b) Mr. Starnes believes that less than $75\%$ of the students at his school completed their math homework last night. The math teachers inspect the homework assignments from a random sample of students at the school to help Mr. Starnes test his claim.

- Check conditions for carrying out a test about a population proportion or mean.

- Interpret $P$-values in context.

 Eye black Athletes performing in bright sunlight often smear black grease under their eyes to reduce glare. Does cye black work? In one experiment, 16 randomly selected student subjects took a test of sensitivity to contrast after 3 hours facing into bright sun, both with and without eye black. Here are the differences in sensitivity, with eye black mines without eye black:

We want to know whether cye black increases sensitivity an the average.

(a) State hypotheses, Be sure to define the parameter.

(b) Check conditions for carrying out a significance test.

(c) The$P value$ of the test is $0.047$. Interpet this value in context.

- Interpret a Type l error and a Type ll error in context, and give the consequences of each.

- Understand the relationsonship between the significance level at a test P(Type li error), and power.

 Strong chairs? A company that manufactures classroom chairs for high school students claims that the mean breaking strength of the chairs that they make is $300$ pounds. One of the chairs collapsed beneath a $220$-pound student last week. You wonder whether the manufacturer is exaggerating the breaking strength of the chairs.

(a) State null and alternative hypotheses in words and symbols.

(b) Describe a Type I error and a Type II error in this situation, and give the consequences 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 this test to detect $\mu =294$ is $0.71$. Explain what this means to someone who knows little statistics.

(e) Explain two ways that you could increase the power of the test from (d).

- If conditions are met, conduct a significance test about a population proportion.

 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. Of these, $43$ get the flu.

(a) Do these data provide convincing evidence to support the company's claim? Perform an appropriate test to support your answer.

(b) Which kind of mistake - a Type I error or a Type II error-could you have made in (a)? Explain.

(c) From the company's point of view, would a Type I error or Type Il error be more serious? Why?

Roulette an American roulette wheel has $18$ red slots among its $38$ slots.In a random sample of $50$pins,the ball lands in a red slot $31$ times.

(a) Do the data give convincing evidence that the wheel is unfair? Carry out an appropriate test at the$\alpha =0.05$ significance level to help answer this question.

(b) The casino manager uses your data to produce a $99\%$ confidence interval for $p$ and gets $\left(0.44,0.80\right)$. He says that this interval provides convincing evidence that the wheel is fair. How do you respond?

- If conditions are met, conduct a one-sample \(t\) test about a population mean \(\mu\).

 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 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 omate font named Gigi. Here are their times, in seconds:

Do these data provide good evidence that the mean reading time for Gigi is greater than  $22$seconds? Carry out an appropriate test to help you answer this question.

- Use a confidence interval to draw a conclusion for a two-sided test about a population mean.

Radon detectors Radon is a colorless, odorless gas that is naturally released by rocks and soils and may concentrate in tightly closed houses. Because radon is slightly radioactive, there is some concern that it may be a health hazard. Radon detectors are sold to homeowners worried about this risk, but the detectors may be inaccurate. University researchers placed a random sample of $11$ detectors in a chamber where they were exposed to $105$ picocuries per liter of radon over $3$ days. A graph of the radon readings from the $11$ detectors shows no strong skewness or outliers. The Minitab output below shows the results of a one-sample t interval. Is there significant evidence at the $10\%$ level that the mean reading $\mu$ differs from the true value $105$? Give appropriate evidence to support your answer.

Better barley Does drying barley seeds in a kiln increase the yield of barley? A famous experiment by William S. Gosset (who discovered the t distributions) investigated this question. Eleven pairs of adjacent plots were marked out in a large field. For each pair, regular barley seeds were planted in one plot and kiln-dried seeds were planted in the other. The following table displays the data on yield (lb/acre).

(a) How can the Random condition be satisfied in this study? 

(b) Perform an appropriate test to help answer the research question. Assume that the Random condition is met. What conclusion would you draw? 

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 

a).$H_0:\hat{p}=0.5;H_a:\hat{p}>0.5$

(b) $H_0:p=0.5;H_a:p>0.5$

(c) $H_0:p=0.5;H_a:p<0.5$

(d) $H_0:p=0.5;H_a:p\ne 0.5$

(e) $H_0:p>0.5;H_a:p=0.5$

You are thinking of conducting a one-sample t-test about a population mean M using a $0.05$ significance level. You suspect that the distribution of the population is not Normal and may be moderately skewed. Which of the following statements is correct? 

(a) You should not carry out the test because the population does not have a Normal distribution. 

(b) You can safely carry out the test if your sample size is large and there are no outliers. 

(c) You can safely carry out the test if there are no outliers, regardless of the sample size. 

(d) You can carry out the test only if the population standard deviation is known. 

(e) The $t$ procedures are robust—you can u 

To determine the reliability of experts who interpret lie detector tests in criminal investigations, a random sample of $280$ such cases was studied. The results were 

(a) $15/280$.

(b) $9/280$.

(c) $15/140$.

(d) $9/140$.

(e) $15/146$.

A significance test allows you to reject a null hypothesis $H_0$ in favor of an alternative $H_a$ at the $5\%$ significance level. What can you say about significance at the $1\%$ level? 

(a) $H_0$ can be rejected at the $1\%$ significance level. 

(b) There is insufficient evidence to reject $H_0$ at the $1\%$ significance level. 

c) There is sufficient evidence to accept $H_0$ at the $1\%$ significance level. 

(d) Ha can be rejected at the $1\%$ significance level. 

(e) The answer can’t be determined from the information given 

A random sample of $100$ likely voters in a small city produced $59$ voters in favor of Candidate A. The observed value of the test statistic for testing the null hypothesis $H_0 : p=0.5$ versus the alternative hypothesis $H_a : p=0.5$ is

$\text{ (a) }z=\frac{0.59-0.5}{\sqrt{\frac{0.59(0.41)}{100}}}$

(b). $z=\frac{0.59-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}$

(c). $z=\frac{0.5-0.59}{\sqrt{\frac{0.59(0.41)}{100}}}$

(d). $z=\frac{0.5-0.59}{\sqrt{\frac{0.5(0.5)}{100}}}$

(e). $t=\frac{0.59-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}$