Estimating with Confidence

How much does the fat content of Brand $X$ hot dogs vary? To find out, researchers measured the fat content (in grams) of a random sample of $10$ Brand X hot dogs. A $95\%$ confidence interval for the population standard deviation $S$ is $2.84$ to $7.55$ .

1. Interpret the confidence interval. 

Interpret the confidence level. 

True or false: The interval from $2.84$ to $7.55$ has a $95\%$ chance of containing the actual population standard deviation $S$. Justify your answer. 

Got shoes? How many pairs of shoes, on average, do female teens have? To find out, an AP Statistics class conducted a survey. They selected an SRS of $20$ female students from their school. Then they recorded the number of pairs of shoes that each student reported having. Here are the data:

Got shoes? The class in Exercise 1 wants to estimate the variability in the number of pairs of shoes that female students have by estimating the population variance ${\sigma }^2$. 

Going to the prom Tonya wants to estimate what proportion of the seniors in her school plan to attend the prom. She interviews an SRS of $50$ of the $750$ seniors in her school and finds that $36$ plan to go to the prom 

Reporting cheating What proportion of students are willing to report cheating by other students? A student project put this question to an SRS of $172$ undergraduates at a large university: “You witness two students cheating on a quiz. Do you go to the professor?” Only 19 answered “Yes.”$3$

5. NAEP scores Young people have a better chance of full-time employment and good wages if they are good with numbers. How strong are the quantitative skills of young Americans of working age? One source
of data is the National Assessment of Educational Progress (NAEP) Young Adult Literacy Assessment Survey, which is based on a nationwide probability sample of households. The NAEP survey includes a
short test of quantitative skills, covering mainly basic arithmetic and the ability to apply it to realistic problems. Scores on the test range from $0$ to $500$. For example, a person who scores $233$ can add the amounts of two checks appearing on a bank deposit slip; someone scoring $325$ can determine the price of a meal from a menu; a person scoring $375$ can
transform a price in cents per ounce into dollars per pound. Suppose that you give the NAEP test to an SRS of $840$ people from a large population in which the scores have mean $280$ and standard deviation$\sigma =60$. The mean $\overline{x}$ of the $840$ scores will vary if you take repeated samples.
(a) Describe the shape, center, and spread of the sampling distribution of $\overline{x}$.
(b) Sketch the sampling distribution of $\overline{x}$. Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the$68–95–99.7$ rule, about $95\%$ of all values of $\overline{x}$ lie within a distance $m$ of the mean of the sampling distribution. What is $m$? Shade the region on the axis of your sketch that is within m of the mean.

(d) Whenever $\overline{x}$ falls in the region you shaded, the population mean $\mu$ lies in the confidence interval$\overline{x}\pm m$. For what percent of all possible samples does the interval capture $\mu$?

6. Auto emissions Oxides of nitrogen (called $NOX$for short) emitted by cars and trucks are important contributors to air pollution. The amount of $NOX$
emitted by a particular model varies from vehicle to vehicle. For one light-truck model, $NOX$ emissions vary with mean $\mu$ that is unknown and standard deviation $\sigma = 0.4$ gram per mile. You test an $SRS$ of $50$ of these trucks. The sample mean $NOX$ level $\overline{x}$ estimates the unknown $\mu$. You will get different values of $\overline{x}$ if you repeat your sampling.

(a) Describe the shape, center, and spread of the sampling distribution of $\overline{x}$.
(b) Sketch the sampling distribution of $\overline{x}$. Mark its mean and the values one, two, and three standard deviations on either side of the mean.
(c) According to the $68–95–99.7$ rule, about $95\%$ of all values of $\overline{x}$ lie within a distance $m$ of the mean of the sampling distribution. What is $m$? Shade the region on the axis of your sketch that is within $m$ of the mean.
(d) Whenever $\overline{x}$ falls in the region you shaded, the unknown population mean $\mu$ lies in the confidence interval $\overline{x}\pm m$. For what percent of all possible samples does the interval capture $\mu$?

NAEP scores Refer to Exercise 5. Below your sketch, choose one value of $x$ inside the shaded region and draw its corresponding confidence interval. Do the same for one value of $x$ outside the shaded region. What is the most important difference between these intervals? 

Auto emissions Refer to Exercise 6. Below your sketch, choose one value of $x$ inside the shaded region and draw its corresponding confidence interval. Do the same for one value of $x$ outside the shaded region. What is the most important difference between these intervals? (Use Figure 8.5, on page 474, as a model for your drawing.) 

How confident? The figure below shows the result of taking $25$ SRSs from a Normal population and constructing a confidence interval for each sample. Which confidence level—$80\%, 90\%, 95\%$, or $99\%$— do you think was used? Explain

How confident? The figure at top right shows the result of taking $25$ SRSs from a Normal population and constructing a confidence interval for each sample. Which confidence level—$80\%, 90\%, 95\%$, or $99\%$—do you think was used? Explain.

Prayer in school A New York Times/CBS News Poll asked the question, “Do you favor an amendment to the Constitution that would permit organized prayer in public schools?” Sixty-six percent of the sample answered “Yes.” The article describing the poll says that it “is based on telephone interviews conducted from Sept. 13 to Sept. 18 with $1664$ adults around the United States, excluding Alaska and Hawaii. . . . The telephone numbers were formed by random digits, thus permitting access to both listed and unlisted residential numbers.” The article gives the margin of error for a $95\%$ confidence level as $3$ percentage points.

(a) Explain what the margin of error means to someone who knows little statistics. 

(b) State and interpret the 95% confidence interval. 

(c) Interpret the confidence level. 

Losing weight A Gallup Poll in November $2008$ found that $59\%$ of the people in its sample said “Yes” when asked, “Would you like to lose weight?” Gallup announced: “For results based on the total sample of national adults, one can say with 95% confidence that the margin of (sampling) error is $\pm 3$ percentage points.”

 (a) Explain what the margin of error means in this setting. 

(b) State and interpret the $95\%$ confidence interval. 

(c) Interpret the confidence level. 

The news article goes on to say: “The theoretical errors do not take into account a margin of additional error resulting from the various practical difficulties in taking any survey of public opinion.” List some of the “practical difficulties” that may cause errors in addition to the $\pm 3$ percentage point margin of error. Pay particular attention to the news article’s description of the sampling method. 

Refer to Exercise $12$. As Gallup indicates, the $3$ percentage point margin of error for this poll includes only sampling variability (what they call “sampling error”). What other potential sources of error (Gallup calls these “non-sampling errors”) could affect the accuracy of the $59\%$ estimate? 

The AP Statistics class in Exercise $1$ also asked an SRS of $20$ boys at their school how many shoes they have. A $95\%$ confidence interval for the difference in the population means (girls – boys) is $10.9$ to $26.5$. Interpret the confidence interval and the confidence level. 

Many teens have posted profiles on sites such as Facebook and My Space. A sample survey asked random samples of teens with online profiles if they included false information in their profiles. Of $170$ younger teens (ages $12$ to $14$) polled, $117$ said “Yes.” Of $317$ older teens (ages $15$ to $17$) polled, $152$ said “Yes.”$6$ A $95\%$ confidence interval for the difference in the population proportions (younger teens – older teens) is $0.120$ to $0.297$. Interpret the confidence interval and the confidence level 

17. Explaining confidence A $95\%$ confidence interval for the mean body mass index $\left(BMI\right)$ of young American women is $26.8\pm 0.6$. Discuss whether each of the following explanations is correct.
(a) We are confident that $95\%$ of all young women have $BMI$ between $26.2$ and $27.4$.
(b) We are $95\%$ confident that future samples of young women will have mean $BMI$ between $26.2$ and $27.4$.
(c) Any value from $26.2$ to $27.4$ is believable as the true mean $BMI$ of young American women.
(d) In $95\%$ of all possible samples, the population mean $BMI$ will be between $26.2$ and $27.4$.
(e) The mean $BMI$ of young American women cannot be $28$.

18. Explaining confidence The admissions director from Big City University found that (107.8, 116.2) is a 95% confidence interval for the mean IQ score of all freshmen. Comment on whether or not each of the following explanations is correct.
(a) There is a $95\%$ probability that the interval from $107.8$ to $116.2$ contains $\mu$.
(b) There is a $95\%$ chance that the interval $(107.8, 116.2$) contains $\overline{x}$.
(c) This interval was constructed using a method that produces intervals that capture the true mean in $95\%$ of all possible samples.
(d) $95\%$ of all possible samples will contain the interval $(107.8, 116.2)$.

(e) The probability that the interval $(107.8, 116.2)$ captures $\mu$ is either $0$ or $1$, but we don’t know which.

Explain briefly why each of the three conditions—Random, Normal, and Independent—is important when constructing a confidence interval. 

An online poll posed the following question: 

It is now possible for school students to log on to Internet sites and download homework. Everything from book reports to doctoral dissertations can be downloaded free or for a fee. Do you believe that giving a student who is caught plagiarizing an F for their assignment is the right punishment? 

Of the $20,125$ people who responded, $14,793$ clicked “Yes.” That’s $73.5\%$ of the sample. Based on this sample, a $95\%$ confidence interval for the percent of the population who would say “Yes” is $73.5\%$ $\pm 0.61$. Which of the three inference conditions is violated? Why is this confidence interval worthless? 

A researcher plans to use a random sample of $n=500$ families to estimate the mean monthly family income for a large population. A $99\%$ confidence interval based on the sample would be ______ than a 90% confidence interval.

(a) narrower and would involve a larger risk of being incorrect 

(b) wider and would involve a smaller risk of being incorrect 

(c) narrower and would involve a smaller risk of being incorrect 

(d) wider and would involve a larger risk of being incorrect 

(e) wider, but it cannot be determined whether the risk of being incorrect would be larger or smaller 

In a poll, 

I. Some people refused to answer questions. 

II. People without telephones could not be in the sample. 

III. Some people never answered the phone in several calls. 

Which of these sources is included in the $\pm 2\%$ margin of error announced for the poll? 

(a) I only 

(b) II only 

(c) III only

(d) I, II, and III 

 (e) None of these  

You have measured the systolic blood pressure of an SRS of $25$ company employees. A $95\%$ confidence interval for the mean systolic blood pressure for the employees of this company is $(122, 138)$. Which of the following statements gives a valid interpretation of this interval?

(a) $95\%$ of the sample employees have systolic blood pressure between $122 and 138$.

(b) $95\%$ of the population of employees have systolic blood pressure between $122 and 138$.

(c) If the procedure were repeated many times, $95\%$ of the resulting confidence intervals would contain the population mean systolic blood pressure.

(d) The probability that the population mean blood pressure is between $122 and 138$ is 0.95.

(e) If the procedure were repeated many times, $95\%$ of the sample means would be between $122 and 138$. 

A polling organization announces that the proportion of American voters who favor congressional term limits is  $64\%$, with a $95\%$ confidence margin of error of $3\%$. If the opinion poll had announced the margin of error for 80% confidence rather than $95\%$ confidence, this margin of error would be

(a) $3\%$, because the same sample is used.

(b) less than $3\%$, because we require less confidence.

(c) less than $3\%$, because the sample size is smaller.

(d) greater than $3\%$, because we require less confidence.

(e) greater than $3\%$, because the sample size is smaller. 

Power lines and cancer $(4.2,4.3)$ Does living near power lines cause leukemia in children? The

National Cancer Institute spent 5 years and $\$5$ million gathering data on this question. The researchers compared $638$ children who had leukemia with$620$ who did not. They went into the homes and actually measured the magnetic fields in children's bedrooms, in other rooms, and at the front door. They recorded facts about power lines near the family home and also near the mother's residence when she was pregnant. Result: no connection between leukemia and exposure to magnetic fields of the kind produced by power lines was found.

(a) Was this an observational study or an experiment? Justify your answer.

(b) Does this study show that living near power lines doesn't cause cancer? Explain.

26.  Sisters and brothers (3.1, 3.2) How strongly do physical characteristics of sisters and brothers correlate? Here are data on the heights (in inches)
of 11 adult pairs:

(a) Construct a scatterplot using brother’s height as the explanatory variable. Describe what you see.
(b) Use your calculator to compute the least-squares regression line for predicting sister’s height from brother’s height. Interpret the slope in context.
(c) Damien is $70$ inches tall. Predict the height of his sister Tonya.
(d) Do you expect your prediction in (c) to be very accurate? Give appropriate evidence to support your answer.

In each of the following settings, check whether the conditions for calculating a confidence interval for the population proportion p are met.

1. An AP Statistics class at a large high school conducts a survey. They ask the first $100$ students to arrive at school one morning whether or not they slept at least 8 hours the night before. Only 17 students say "Yes."

2. A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt.

In each of the following settings, check whether the conditions for calculating a confidence interval for the population proportion p are met.

1. An AP Statistics class at a large high school conducts a survey. They ask the first 100 students to arrive at school one morning whether or not they slept at least 8 hours the night before. Only 17 students say "Yes."

2. A quality control inspector takes a random sample of 25 bags of potato chips from the thousands of bags filled in an hour. Of the bags selected, 3 had too much salt.

Alcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of $10,904$ randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems." The researchers defined "frequent binge drinking" as having five or more drinks in row three or more times in the past two weeks. According to this definition, $2486$ students were classified as frequent binge drinkers.

1. Identify the population and the parameter of interest.

2. Check conditions for constructing a confidence interval for the parameter.

3. Find the critical value for a 99 \% confidence interval. Show your method. Then calculate the interval.

4. Interpret the interval in context. 

Alcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of 10,904 randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems.9 The researchers defined “frequent binge drinking” as having five or more drinks in row three or more times in the past two weeks. According to this definition, 2486 students were classified as frequent binge drinkers

Check conditions for constructing a confidence interval for the parameter 

Alcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of 10,904 randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems.9 The researchers defined “frequent binge drinking” as having five or more drinks in row three or more times in the past two weeks. According to this definition, 2486 students were classified as frequent binge drinkers. 

Find the critical value for a 99% confidence interval. Show your method. Then calculate the interval. 

 Alcohol abuse has been described by college presidents as the number one problem on campus, and it is an important cause of death in young adults. How common is it? A survey of 10,904 randomly selected U.S. college students collected information on drinking behavior and alcohol-related problems.9 The researchers defined “frequent binge drinking” as having five or more drinks in row three or more times in the past two weeks. According to this definition, 2486 students were classified as frequent binge drinkers. 

 Interpret the interval in context. Putting It All Together 

 The company’s customer satisfaction survey. 

 In the company’s prior-year survey, 80% of customers surveyed said they were “satisfied” or “very satisfied.” Using this value as a guess for pˆ, find the sample size needed for a margin of error of 3% at a 95% confidence level. 

1. In the company’s prior-year survey, 80% of customers surveyed said they were “satisfied” or “very satisfied.” Using this value as a guess for pˆ, find the sample size needed for a margin of error of 3% at a 95% confidence level.

What if the company president demands 99% confidence instead? Determine how this would affect your answer to Question 1. 

Rating dorm food Latoya wants to estimate what proportion of the seniors at her high school like the cafeteria food. She interviews an SRS of $50$ of the $175$ seniors living in the dormitory. She finds that $14$ think the cafeteria food is good.

High tuition costs Glenn wonders what proportion of the students at his school think that tuition is too high. He interviews an SRS of $50$ of the $2400$ students at his college. Thirty-eight of those interviewed think tuition is too high.

AIDS and risk factors In the National AIDS

Behavioral Surveys sample of $2673$adult heterosexuals, $0.2\%$ had both received a blood transfusion and had a sexual partner from a group at high risk of A I D S. We want to estimate the proportion $p$in the population who share these two risk factors.

Whelks and mussels The small round holes you often see in sea shells were drilled by other sea creatures, who ate the former dwellers of the shells. Whelks often drill into mussels, but this behavior appears to be more or less common in different locations. Researchers collected whelk eggs from the coast of Oregon, raised the whelks in the laboratory, then put each whelk in a container with some delicious mussels. Only $9 of 98$ whelks drilled into a mussel.  The researchers want to estimate the proportion $p$ of Oregon whelks that will spontaneously drill into mussels.

98% Confidence find $z^{*}$ for a $98\%$confidence interval using table A or tour calculator. show your method.

$93\%$ confidence Find $z* for a 93 \%$confidence interval using Table A or your calculator. Show your method.

33. Going to the prom Tonya wants to estimate what proportion of her school’s seniors plan to attend the prom. She interviews an $SRS$ of $50$ of the $750$ seniors in her school and finds that $36$ plan to go to the prom.
(a) Identify the population and parameter of interest.
(b) Check conditions for constructing a confidence interval for the parameter.

(c) Construct a 90% confidence interval for p. Show your method.
(d) Interpret the interval in context.

34. Reporting cheating What proportion of students are willing to report cheating by other students? A student project put this question to an SRS of $172$ undergraduates at a large university: “You witness two students cheating on a quiz. Do you go to the professor?” Only$19$ answered “Yes.”
(a) Identify the population and parameter of interest.
(b) Check conditions for constructing a confidence interval for the parameter.
(c) Construct a $99\%$ confidence interval for $p$. Show your method.
(d) Interpret the interval in context.

Abstain from drinking In a Harvard School of Public Health survey, $2105$ of $10,904$ randomly selected U.S. college students were classified as abstainers (nondrinkers).

(a) Construct and interpret a $99\%$ confidence interval for$p$. Follow the four-step process.

(b) A newspaper article claims that $25\%$ of U.S. college students are nondrinkers. Use your result from (a) to comment on this claim.

Teens' online profiles Over half of all American teens (ages 12 to 17 years) have an online profile, mainly on Facebook. A random sample of 487 teens with profiles found that 385 included photos of themselves. ${}^{13}$

(a) Construct and interpret a $95\%$confidence interval for $p$. Follow the four-step process.

(b) Is it plausible that the true proportion of American teens with profiles who have posted photos of themselves is $0.75 ?$ Use your result from part (a) to support your answer.

Abstain from drinking Describe a possible source of error that is not included in the margin of error for the $99\%$ confidence interval in Exercise $35$

Teens" online profiles Describe a possible source of error that is not included in the margin of error for the $95\%$ confidence interval in Exercise 36

How common is SAT coaching? A random sample of students who took the SAT college entrance examination twice found that $427$of the respondents had paid for coaching courses and that the remaining $2733$ had not. ${}^{1+}$ Construct and interpret a $99\%$ confidence interval for the proportion of coaching among students who retake the SAT. Follow the four-step process.