Estimating with Confidence

2K10 begins In January $2010$, a Gallup Poll asked a random sample of adults, "In general, are you satisfied or dissatisfied with the way things are going in the United States at this time?" In all, $256$ said that they were sitisfied and the remaining $769$ said they were not. Construct and interpret a$90\%$ confidence interval for the proportion of adults who are satisfied with how things are going. Follow the four-step process.

A TV poll A television news program conducts a call-in poll about a proposed city ban on handgun ownership. Of the $2372$ calls, $1921$ oppose the ban. The station, following recommended practice, makes a confidence statement: $"81\%$of the Channel $13$ Pulse Poll sample opposed the ban. We can be $95\%$ confident that the true proportion of citizens opposing a handgun ban is within$1.6\%$ of the sample result."

(a) Is the station's quoted $1.6\%$ margin of error correct? Explain.

(b) Is the station's conclusion justifed? Explain.

Equality for women? Have efforts to promote equality for women gone far enough in the United States? A poll on this issue by the cable network MSNBC contacted $1019$ adults. A newspaper article about the poll said, “Results have a margin of sampling error of plus or minus 3 percentage points.”$15$ 

(a)  The news article said that $65$% of men, but only $43$% of women, think that efforts to promote equality have gone far enough. Explain why we do not have enough information to give confidence intervals for men and women separately.

(b)  Would a $95$% confidence interval for women alone have a margin of error less than $0.03$, about equal to $0.03$, or greater than $0.03$? Why? (You see that the news article’s statement about the margin of error for poll results is a bit misleading.) 

Can you taste PTC? PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste PTC is inherited. About $75$% of Italians can taste PTC, for example. You want to estimate the proportion of Americans who have at least one Italian grandparent and who can taste PTC.

(a)  How large a sample must you test to estimate the proportion of PTC tasters within $0.04$ with $90$% confidence? Answer this question using the $75$% estimate as to the guessed value for $\stackrel{\wedge }{p}$.

(b)  Answer the question in part (a) again, but this time use the conservative guess $\stackrel{\wedge }{p}=0.5$. By how much do the two sample sizes differ?

School vouchers A national opinion poll found that $44$% of all American adults agree that parents should be given vouchers that are good for education at any public or private school of their choice. The result was based on a small sample.

(a)  How large an SRS is required to obtain a margin of error of $0.03$ (that is, $\pm 3\%$) in a $99$% confidence interval? Answer this question using the previous poll’s result as the guessed value for $\stackrel{\wedge }{p}$.

(b)  Answer the question in part (a) again, but this time use the conservative guess $\stackrel{\wedge }{p}=0.5$. By how much do the two sample sizes differ? 

Election polling Gloria Chavez and Ronald Flynn are the candidates for mayor in a large city. We want to estimate the proportion $p$ of all registered voters in the city who plan to vote for Chavez with $95$% confidence and a margin of error no greater than $0.03$. How large a random sample do we need? Show your work. 

Starting a nightclub A college student organization wants to start a nightclub for students under the age of $21$. To assess support for this proposal, they will select an SRS of students and ask each respondent if he or she would patronize this type of establishment. They expect that about $70$% of the student body would respond favorably. What sample size is required to obtain a $90$% confidence interval with an approximate margin of error of $0.04$? Show your work

Teens and their TV sets According to a Gallup Poll report, $64$% of teens aged $13$ to $17$ have TVs in their rooms. Here is part of the footnote to this report:  

These results are based on telephone interviews with a randomly selected national sample of $1028$ teenagers in the Gallup Poll Panel of households, aged $13$ to $17$. For results based on this sample, one can say . . . that the maximum error attributable to sampling and other random effects is $\pm 3$ percentage points. In addition to sampling error, question-wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls.$16$

(a)  We omitted the confidence level from the footnote. Use what you have learned to determine the confidence level, assuming that Gallup took an SRS. 

(b)  Give an example of a “practical difficulty” that could lead to biased results for this survey.

Gambling and the NCAA Gambling is an issue of great concern to those involved in college athletics. Because of this concern, the National Collegiate Athletic Association (NCAA) surveyed randomly selected student-athletes concerning their gambling-related behaviors.$17$ Of the $5594$ Division I male athletes in the survey, $3547$ reported participation in some gambling behavior. This includes playing cards, betting on games of skill, buying lottery tickets, betting on sports, and similar activities. A report of this study cited a $1$% margin of error.

(a)  The confidence level was not stated in the report. Use what you have learned to find the confidence level, assuming that the NCAA took an SRS.

(b)  The study was designed to protect the anonymity of the student-athletes who responded. As a result, it was not possible to calculate the number of students who were asked to respond but did not. How does this fact affect the way that you interpret the results?

49. A Gallup Poll found that only $28\%$ of American adults expect to inherit money or valuable possessions from a relative. The poll’s margin of error was $\pm 3$ percentage points at a $95\%$ confidence level. This means that
(a) the poll used a method that gets an answer within $3\%$ of the truth about the population $95\%$ of the time.
(b) the percent of all adults who expect an inheritance is between $25\%$ and $31\%$.
(c) if Gallup takes another poll on this issue, the results of the second poll will lie between $25\%$ and $31\%$.
(d) there’s a $95\%$ chance that the percent of all adults who expect an inheritance is between $25\%$ and $31\%$.
(e) Gallup can be $95\%$ confident that between $25\%$ and $31\%$ of the sample expect an inheritance

50. Most people can roll their tongues, but many can’t. The ability to roll the tongue is genetically determined. Suppose we are interested in determining what proportion of students can roll their tongues. We test a simple random sample of $400$ students and find that $317$ can roll their tongues. The margin of error for a $95\%$ confidence interval for the true proportion of tongue rollers among students is closest to
(a) $0.008$. (c) $0.03$. (e) $0.208$.
(b) $0.02$. (d) $0.04$

51. You want to design a study to estimate the proportion of students at your school who agree with the statement, “The student government is an effective organization for expressing the needs of students to the administration.” You will use a $95\%$ confidence interval, and you would like the margin of error to be $0.05$ or less. The minimum sample size required is
(a) $22$. (b) $271$. (c) $385$. (d) $769$. (e) $1795$.

I collect an SRS of size n from a population and compute a $95\%$ confidence interval for the population proportion. Which of the following would produce a new confidence interval with larger width (larger margin of error) based on these same data?

(a) Use a larger confidence level.

(b) Use a smaller confidence level. 

(c) Increase the sample size. 

(d) Use the same confidence level, but compute the interval n times. Approximately$5\%$ of these intervals will be larger. 

(e) Nothing can guarantee absolutely that you will get a larger interval. One can only say that the chance of obtaining a larger interval is $0.05$. 

Exercises 53 and 54 refer to the following setting. The table below displays the number of accidents at a factory during each hour of a 24-hour shift (1  1 a.m).

53. Accidents happen (1.2, 3.1)
(a) Construct a plot that displays the distribution of the number of accidents effectively.
(b) Construct a plot that shows the relationship between the number of accidents and the time when they occurred.
(c) Describe something that the plot in part (a) tells you about the data that the plot in part (b) does not.
(d) Describe something that the plot in part (b) tells you about the data that the plot in part (a) does not.

Plant managers are concerned that the number of accidents may be significantly higher during the midnight to$8:00 a.m$. shift than during the $4:00 p.m.$ to midnight shift. What would you tell them? Give appropriate statistical evidence to support your conclusion. 

To assess the accuracy of a laboratory scale, a standard weight known to weigh $10$ grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean (this mean is $10$ grams if the scale has no bias). In previous studies, the standard deviation of the scale readings has been about $0.0002$ gram. How many measurements must be averaged to get a margin of error of $0.0001$ with $98\%$ confidence? Show your work.

Use Table B to find the critical value t* that you would use for a confidence interval for a population mean M in each of the following situations. If possible, check your answer with technology. 

(a) A 98% confidence interval based on $n =22$ observations.

 (b) A $90\%$ confidence interval from an SRS of $10$ observations.

(c) A $95\%$ confidence interval from a sample of size $7$.

 Define the parameter of interest. 

What inference method will you use? Check that the conditions for using this procedure are met. 

Construct a $95\%$ confidence interval for$M.$ Show your method 

Interpret your interval in context. 

The body mass index (BMI) of all American young women is believed to follow a Normal distribution with a standard deviation of about $7.5$. How large a sample would be needed to estimate the mean BMI M in this population to within $\pm 1v$ with $99\%$ confidence? Show your work. 

56. The SAT again High school students who take the SAT Math exam a second time generally score higher than on their first try. Past data suggest that the score increase has a standard deviation of about $50$ points.
How large a sample of high school students would be needed to estimate the mean change in SAT score to within $2$ points with $95\%$ confidence? Show your work.

57. Critical values What critical value t* from Table B would you use for a confidence interval for the population mean in each of the following situations?
(a) A $95\%$ confidence interval based on  $n=10$ observations.
(b) A $99\%$ confidence interval from an SRS of $20$ observations.

58. Critical values What critical value t* from Table $B$ should be used for a confidence interval for the population mean in each of the following situations?
(a) A $90\%$confidence interval based on n  $n=12$ observations.
(b) A $95\%$ confidence interval from an SRS of$30$ observations.

59.Blood pressure A medical study finds that $\overline{x}=114.9$ and $S_x= 21.88$ for the seated systolic blood pressure of the $27$ members of one treatment group. What is the standard error of the mean? Interpret this value in context.

60. Travel time to work A study of commuting times reports the travel times to work of a random sample of $20$ employed adults in New York State. The mean is $\overline{x}=31.25$ minutes, and the standard deviation is $s_x= 21.88$minutes. What is the standard error of the mean? Interpret this value in context.

61. Willows in Yellowstone Writers in some fields summarize data by giving $\overline{x}$ and its standard error rather than $\overline{x}$ and $s_x$. Biologists studying willow trees in Yellowstone National Park reported their results in a table with columns labeled $\overline{x}+SE$. The table entry for the heights of willow trees (in centimeters) in one region of the park was $61.55 \pm 19.03.$ The researchers measured a total of $23$ trees.
(a) Find the sample standard deviation $s_x$ for these measurements. Show your work.
(b) Explain why the given interval is not a confidence interval for the mean height of willow trees in this region of the park.

62. Blink When two lights close together blink alternately, we “see” one light moving back and forth if the time between blinks is short. What is the
longest interval of time between blinks that preserves the illusion of motion? Ask subjects to turn a knob that slows the blinking until they “see” two lights rather than one light moving. A report gives the results in the form “mean plus or minus the standard error of the mean.” Data for $12$ subjects are summarized as $251\pm 45$ (in milliseconds).
(a) Find the sample standard deviation $s_x$ for these measurements. Show your work.
(b) Explain why the interval $251\pm 45$ is not a confidence interval.

63. Give it some gas! Computers in some vehicles calculate various quantities related to performance. One of these is fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle
equipped in this way, the miles per gallon were recorded each time the gas tank was filled and the computer was then reset. Here are the mpg values for a random sample of $20$ of these records:

$15.8 13.6 15.6 19.1 22.4 15.6 22.5 17.2 19.4 22.6$
$19.4 18.0 14.6 18.7 21.0 14.8 22.6 21.5 14.3 20.9$
Construct and interpret a $95\%$ confidence interval for the mean fuel efficiency M for this vehicle.

64. Vitamin C content Several years ago, the U.S. Agency for International Development provided $238,300$ metric tons of corn-soy blend (CSB) for
emergency relief in countries throughout the world. CSB is a highly nutritious, low-cost fortified food. As part of a study to evaluate appropriate vitamin C levels in this food, measurements were taken on samples of CSB produced in a factory. The following data are the amounts of vitamin C,

measured in milligrams per $100$ grams (mg/100 g) of blend, for a random sample of size$8$ from one production run:
$26 31 23 22 11 22 14 31$
Construct and interpret a$95\%$ confidence interval for the mean amount of vitamin C M in the CSB from this production run.  

65. Critical value What critical value$t*$ from Table $B$would you use for a $99\%$ confidence interval for the population mean based on an SRS of size $58$? If possible, use technology to find a more accurate value of $t*$. What advantage does the more accurate $df$provide?

What critical value $t^{\star }$ from Table B would you use for a $90\%$ confidence interval for the population mean based on an SRS of size $77$? If possible, use technology to find a more accurate value of $t^{\star }$. What advantage does the more accurate df provide?

Critical value What critical value $t^{*}$ from Table B would you use for a $90\%$ confidence interval for the population mean based on an SRS of size $77$? If possible, use technology to find a more accurate value of . What advantage does the more accurate df provide? 

Breast-feeding mothers secrete calcium into their milk. Some of the calcium may come from their bones, so mothers may lose bone mineral. Researchers measured the percent change in bone mineral content (BMC) of the spines of $47$ randomly selected mothers during three months of breastfeeding. The mean change in BMC was $-3.587\%$ and the standard deviation was $2.506\%$.

(a) Construct and interpret a $99\%$ confidence interval to estimate the mean percent change in BMC in the population.

(b) Based on your interval from (a), do these data give good evidence that on average nursing mothers lose bone mineral? Explain. 

Bone loss by nursing mothers.: Breastfeeding mothers secrete calcium into their milk. Some of the calcium may come from their bones, so mothers may lose bone minerals. Researchers measured the per cent change in bone mineral content (BMC) of the spines of $47$ randomly selected mothers during three months of breastfeeding. The mean change in BMC was $3.587\%$ and the standard deviation was $2.506\%$.

(a) Construct and interpret a 99% confidence interval to estimate the mean per cent change in BMC in the population. 

(b) Based on your interval from (a), do these data give good evidence that on average nursing mothers lose bone minerals? Explain 

Hallux abducto valgus (call it HAV) is a deformation of the big toe that is fairly uncommon in youth and often requires surgery. Doctors used $X$-rays to measure the angle (in degrees) of deformity in a random sample of patients under the age of $21$ who came to a medical center for surgery to correct HAV. The angle is a measure of the seriousness of the deformity. For these $21$ patients, the mean HAV angle was $24.76$ degrees and the standard deviation was $6.34$ degrees. A dot plot of the data revealed no outliers or strong skewness.

(a) Construct and interpret a $90\%$ confidence interval for the mean HAV angle in the population of all such patients.

(b) Researchers omitted one patient with an HAV angle of $50$ degrees from the analysis due to a measurement issue. What effect would including this outlier have on the confidence interval in (a)? Justify your answer. 

Researchers were interested in comparing two methods for estimating tire wear. The first method used the amount of weight lost by a tire. The second method used the amount of wear in the grooves of the tire. A random sample of $16$ tires was obtained. Both methods were used to estimate the total distance traveled by each tire. The table below provides the two estimates (in thousands of miles) for each tire.

(a) Construct and interpret a $95\%$ confidence interval for the mean difference $\mu$ in the estimates from these two methods in the population of tires.

(b) Does your interval in part (a) give convincing evidence of a difference in the two methods of estimating tire wear? Justify your answer. 

$\begin{array}{ccccccc}\text{ Tire }& \text{ Weight }& \text{ Groove }& & \text{ Tire }& \text{ Weight }& \text{ Groove }\\ 1& 45.9& 35.7& & 9& 30.4& 23.1\\ 2& 41.9& 39.2& & 10& 27.3& 23.7\\ 3& 37.5& 31.1& & 11& 20.4& 20.9\\ 4& 33.4& 28.1& & 12& 24.5& 16.1\\ 5& 31.0& 24.0& & 13& 20.9& 19.9\\ 6& 30.5& 28.7& & 14& 18.9& 15.2\\ 7& 30.9& 25.9& & 15& 13.7& 11.5\\ 8& 31.9& 23.3& & 16& 11.4& 11.2\\ & & & & & & \end{array}$

Trace metals found in wells affect the taste of drinking water, and high concentrations can pose a health risk. Researchers measured the concentration of zinc (in milligrams/liter) near the top and the bottom of $10$ randomly selected wells in a large region. The data are provided in the table below. 

(a) Construct and interpret a $95\%$ confidence interval for the mean difference $\mu$ in the zinc concentrations from these two locations in the wells.

(b) Does your interval in part (a) give convincing evidence of a difference in zinc concentrations at the top and bottom of wells in the region? Justify your answer. 

How heavy a load (pounds) is needed to pull apart pieces of Douglas fir $4$ inches long and $1.5$ inches square? A random sample of $20$ similar pieces of Douglas fir from a large batch was selected for a science class. The Fathom boxplot below shows the class’s data. Explain why it would not be wise to use a one-sample t interval to estimate the population mean $\mu$.

Velvetleaf is a particularly annoying weed in cornfields. It produces lots of seeds, and the seeds wait in the soil for years until conditions are right for sprouting. How many seeds do velvetleaf plants produce? The Fathom histogram below shows the counts from $28$ plants that came up in a cornfield when no herbicide was used. Explain why it would not be wise to use a one-sample $t$ interval to estimate the mean number of seeds $\mu$ produced by velvetleaf plants.

In each of the following situations, discuss whether it would be appropriate to construct a one-sample $t$ interval to estimate the population mean.

(a) We collect data from a random sample of adult residents in a state. Our goal is to estimate the overall percentage of adults in the state who are college graduates.

 (b) The coach of a college men’s basketball team records the resting heart rates of the $15$ team members. We use these data to construct a confidence interval for the mean resting heart rate of all male students at this college.

(c) Do teens text more than they call? To find out, an AP Statistics class at a large high school collected data on the number of text messages and calls sent or received by each of $25$ randomly selected students. The Fathom boxplot below displays the difference (texts calls) for each student.

In each of the following situations, discuss whether it would be appropriate to construct a one-sample $t$ interval to estimate the population mean.

(a) We want to estimate the average age at which U.S. presidents have died. So we obtain a list of all U.S. presidents who have died and their ages at death. 

(b) How much time do students spend on the Internet? We collect data from the $32$ members of our AP Statistics class and calculate the mean amount of time that each student spent on the Internet yesterday.

(c) Judy is interested in the reading level of a medical journal. She records the length of a random sample of $100$ words from a multipage article. The Minitab histogram below displays the data.

One reason for using a $t$ distribution instead of the standard Normal curve to find critical values when calculating a level C confidence interval for 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 that you can regard your data as an SRS from the population.

(d) the standard Normal table doesn't include confidence levels at the bottom.

(e) a $z$ critical value will lead to a wider interval than a $t$ critical value.

You have an SRS of $23$ observations from a Normally distributed population. What critical value would you use to obtain a $98\%$ confidence interval for the mean M of the population if S is unknown? 

(a) $2.508$

(b) $2.500$

(c) $2.326$

(d) $2.183$ 

(e) $2.177$

A quality control inspector will measure the salt content (in milligrams) in a random sample of bags of potato chips from an hour of production. which of  the following would result in the smallest margin of error in estimating the mean salt content $M$? 

(a) $90\%$ confidence; $n=25$.

(b) $90\%$ confidence; $n=50$ .

(c) $95\%$ confidence; $n=25$.

(d) $95\%$confidence; $n=50$.

(e) $n=100$ at any confidence level.

Watching TV (6.1, 7.3) Choose a young person (aged $19$ to $25$) at random and ask, “In the past seven days, how many days did you watch television?” Call the response $X$ for short. Here is the probability distribution for $X$.

(a) What is the probability that $X=7$? Justify your answer.

(b) Calculate the mean of the random variable X. Interpret this value in context. 

(c) Suppose that you asked $100$ randomly sele

cted young people (aged $19$ to $25$) to respond to the question and found that the mean $x$ of their responses was $4.96$. Would this result surprise you? Justify your answer.

Scientists collect data on the blood cholesterol levels (milligrams per deciliter of blood) of a random sample of $24$ laboratory rats. A $95\%$ confidence interval for the mean blood cholesterol level $M$ is $80.2$ to $89.8$. Which of the following would cause the most worry about the validity of this interval?

(a) There is a clear outlier in the data. 

(b) A stemplot of the data shows a mild right skew. 

(c) You do not know the population standard deviation $S$. 

(d) The population distribution is not exactly Normal. 

(e) None of these would be a problem because the t procedures are robust. 

Price cuts (4.2) Stores advertise price reductions to attract customers. What type of price cut is most attractive? Experiments with more than one factor allow insight into interactions between the factors. A study of the attractiveness of advertised price discounts had two factors: percent of all foods on sale $(25\%, 50\%, 75\%, or 100\%)$ and whether the discount was stated precisely (as in, for example, $“60\%$ off”) or as a range (as in “$40\%$ to $70\%$ off”). Subjects rated the attractiveness of the sale on a scale of 1 to 7. (a) Describe a completely randomized design using $200$ student subjects. (b) Explain how you would use the partial table of random digits below to assign subjects to treatment groups. Then use your method to select the first $3$ subjects for one of the treatment groups. Show your work clearly on your paper.

 (c) The figure below shows the mean ratings for the eight treatments formed from the two factors. Based on these results, write a careful description of how percent on sale and precise discount versus range of discounts influence the attractiveness of a sale. 

Conditions Martin says that the relative importance of the three conditions for performing inference is, in order from most to least important, Independent, Normal, and Random. Write a brief note to Martin explaining why he is incorrect.