Estimating with Confidence

It’s critical Find the appropriate critical value for constructing a confidence interval in each of the following settings. 

(a) Estimating a population proportion p at a $94\%$ confidence level based on an SRS of size $125$. 

(b) Estimating a population mean M at a $99\%$ confidence level based on an SRS of size $58$. 

 Do you go to church? The Gallup Poll plans to ask a random sample of adults whether they attended a religious service in the last $7$ days. How large a sample would be required to obtain a margin of error of $0.01 in a 99\%$ confidence interval for the population proportion who would say that they attended a religious service? Show your work.

- Construct and interpret a confidence interval for a population proportion.

- Explain how practical issues like nonresponse, under coverage, and response bias can affect the interpretation of a confidence interval.

Running red lights A random digit dialing telephone survey of$880$ drivers asked, "Recalling the last ten traffic lights you drove through, how many of them were red when you entered the intersections?" Of the $880$ respondents, $171$admitted that at least one light had been red.${}^{15}$

(a) Construct and interpret a$95\%$ confidence interval for the population proportion.

(b) Nonresponse is a practical problem for this survey-only $21.6\%$ of calls that reached a live person were completed. Another practical problem is that people may not give truthful answers. What is the likely direction of the bias: do you think more or fewer than $171$of the $880$ respondents really ran a red light? Why? Are these sources of bias included in the margin of error?

 Engine parts Here are measurements (in millimeters) of a critical dimension on an SRS of $16$ of the more than $200$ auto engine crankshafts produced in one day:

$224.120$, $224.001$, $224.017$ ,$223.982,$$223.989,$$223.960,$$224.089,$$223.987,$$223.976,$$223.902,$$223.980,$$224.098,$$224.057,$$223.913,$$223.999$

(a) Construct and interpret a $95\%$ confidence interval for the process mean at the time these crankshafts were produced.

(b) The process mean is supposed to be $\mu =224$ but can drift away from this target during production. Does your interval from part (a) suggest that the process mean has drifted? Explain.

- Determine the sample size required to obtain a level $C$ confidence interval for a population mean with a specified margin of error.

R8.3. Batteries A company that produces AA batteries tests the lifetime of a random sample of 40 batteries using a special device designed to imitate real-world use. Based on the testing, the company makes the following statement: “Our AA batteries last an average of 430 to 470 minutes,
and our confidence in that interval is 95%.
(a) Determine the sample mean and standard deviation.
(b) A reporter translates the statistical announcement into “plain English” as follows: “If you buy one of this company’s AA batteries, there is a 95% chance that it will last between 430 and 470 minutes.” Comment on this interpretation.
(c) Your friend, who has just started studying statistics, claims that if you select 40 more AA batteries at random from those manufactured by this company, there is a 95% probability that the mean life time will fall between 430 and 470 minutes. Do you agree? Explain.
(d) Give a statistically correct interpretation of the confidence interval that could be published in a newspaper report.

R8.4. We love football! A recent Gallup Poll conducted telephone interviews with a random sample of adults aged 18 and older. Data were obtained for $1000$ people. Of these, $37\%$ said that football is their favorite sport to watch
on television.
(a) Define the parameter $p$in this setting. Explain to someone who knows no statistics why we can’t just say that $37\%$of all adults would say that football is their favorite sport to watch on television.
(b) Check conditions for constructing a confidence interval for $p$.
(c) Construct a 95% confidence interval for p.
(d) Interpret the interval in context.

R8.5. Smart kids A school counselor wants to know how smart the students in her school are. She gets funding from the principal to give an IQ test to an SRS of $60$ of the over $1000$ students in the school. The mean IQ score was $114.98$ and the standard deviation was $14.80$.
(a) Describe the parameter $\mu$ in this setting.
(b) Explain why the Normal condition is met in this case.
(c) Construct a $90\%$ confidence interval for the mean IQ score of students at the school.
(d) Interpret your result from part (c) in context.  

Running red lights A random digit dialing telephone survey of 880 drivers asked, “Recalling the last ten traffic lights you drove through, how many of them were red when you entered the intersections?” Of the 880 respondents, 171 admitted that at least one light had been red.

(a) Construct and interpret a $95\%$ confidence interval for the population proportion.

(b) Nonresponse is a practical problem for this survey—only $21.6\%$ of calls that reached a live person were completed. Another practical problem is that people may not give truthful answers. What is the likely direction of the bias: do you think more or fewer than $171$of the $880$ respondents really ran a red light? Why? Are these sources of bias included in the margin of error?

 Good wood? A lab supply company sells pieces of Douglas fir $4$ inches long and $1.5$ inches square for force experiments in science classes. From experience, the strength of these pieces of wood follows a Normal distribution with standard deviation $3000$ pounds. You want to estimate the mean load needed to pull apart these pieces of wood to within $1000$ pounds with $95\%$ confidence. How large a sample is needed? Show your work.

- Understand how the margin of error of a confidence interval changes with the sample size and the level of confidence $C$.

It's about ME Explain how each of the following would affect the margin of error of a confidence interval, if all other things remained the same.

(a) Increasing the confidence level

(b) Quadrupling the sample size

 The Gallup Poll interviews $1600$ people. Of these, $18\%$ say that they jog regularly. The news report adds: "The poll had a margin of error of plus or minus three percentage points at a $95\%$ confidence level." You can safely conclude that

(a)  $95\%$of all Gallup Poll samples like this one give answers within$\pm 3\%$ of the true population value.

(b) the percent of the population who jog is certain to be between $15\% and 21\%$.

(c) $95\%$ of the population jog between$5\% and 21\%$ of the time.

(d) we can be $95\%$ confident that the sample proportion is captured by the confidence interval.

(e) if Gallup took many samples,$95\%$ of them would find that $18\%$ of the people in the sample jog.

 The weights (in pounds) of three adult males are $160,215 and 195$ . The standard error of the mean of these three weights is

(a) $190$

(b) $27.84$

(c) $22.73$

(d) $16.07$

(e) $13.13$

 In preparing to construct a one-sample $t$  interval for a population mean, suppose we are not sure if the population distribution is Normal. In which of the following circumstances would we not be safe constructing the interval based on an SRS of size $24$ from the population?

(a) A stem plot of the data is roughly bell-shaped.

(b) A histogram of the data shows slight skewness.

(c) A stem plot of the data has a large outlier.

(d) The sample standard deviation is large.

(c) The $t$ procedures are robust, so it is always safe.

Many television viewers express doubts about the validity of certain commercials. In an attempt to answer their critics, Timex Group USA wishes to estimate the proportion of consumers who believe what is shown in Timex television commercials. Let  $p$represent the true proportion of consumers who believe what is shown in Timex television commercials. What is the smallest number of consumers that Timex can survey to guarantee a margin of error of $0.05$ or less at a $99\%$confidence level?

(a) $550$

(b) $600$

(c) $650$

(d) $700$

(c) $750$

You want to compute a $90\%$ confidence interval for the mean of a population with unknown population standard deviation. The sample size is $30$ . The value of $t$ you would use for this interval is

(a)$1.645$ (b).$1.699$ (c)$1.697$

(d) $1.96$ (e) $2.045$

 A radio talk show host with a large audience is interested in the proportion $p$of adults in his listening area who think the drinking age should be lowered to eighteen. To find this out, he poses the following question to his listeners: "Do you think that the drinking age should be reduced to eighteen in light of the fact that eighteen-year-olds are eligible for military service?" He asks listeners to phone in and vote "Yes" if they agree the drinking age should be lowered and "No" if not. Of the $100$ people who phoned in, $70$ answered "Yes." Which of the following conditions for inference about a proportion using a confidence interval are violated?

I. The data are a random sample from the population of interest.

II. $n$ is so large that both  $n\overbrace{p} and n\left(1-\overbrace{p}\right)$are at least $10$ .

III. The population is at least $10$times as large as the sample.

(a) I only

(c) III only

(c) I, II, and III

(b) II only

(d) I and II only

A 90% confidence interval for the mean M of a population is computed from a random sample and is found to be 9 $\pm$ 3. Which of the following could be the 95% confidence interval based on the same data?

(a) 9 $\pm$ 1.96 (b) 9 $\pm$ 2 (c) 9 $\pm$ 3 (d) 9 $\pm$ 4

(e) Without knowing the sample size, any of the above answers could be the 95% confidence interval.

A telephone poll of an SRS of 1234 adults found that 62% are generally satisfied with their lives. The announced margin of error for the poll was 3%. Does the margin of

 error account for the fact that some adults do not have telephones?

(a) Yes. The margin of error includes all sources of error in the poll.

(b) Yes. Taking an SRS eliminates any possible bias in estimating the population proportion.

(c) Yes. The margin of error includes under coverage but not nonresponse.

(d) No. The margin of error includes nonresponse but is not under coverage.

(e) No. The margin of error only includes sampling variability.

A Census Bureau report on the income of Americans says that with 90% confidence the median income of all U.S. households in a recent year was \(57,005 with a margin of error of $\pm$\)742. This means that

(a) 90% of all households had incomes in the range \(57,005 $\pm$\)742.

(b) we can be sure that the median income for all households in the country lies in the range \(57,005 $\pm$\)742.

(c) 90% of the households in the sample interviewed by the Census Bureau had incomes in the range \(57,005 $\pm$ \)742.

(d) the Census Bureau got the result \(57,005 $\pm$ \)742 using a method that will cover the true median income 90% of the time when used repeatedly.

(e) 90% of all possible samples of this same size would result in a sample median that falls within \(742 of \)57,005.

The U.S. Forest Service is considering additional restrictions on the number of vehicles allowed to enter Yellowstone National Park. To assess public reaction, the service asks a random sample of 150 visitors if they favour the proposal. Of these, 89 say “Yes.”

(a) Construct and interpret a 99% confidence interval for the proportion of all visitors to Yellowstone who favours the restrictions.

(b) Based on your work in part (a), can the U.S. Forest Service conclude that more than half of visitors to Yellow-stone National Park favour the proposal? Justify your answer.

How many people live in South African households? To find out, we collected data from an SRS of 48 out of the over 700,000 South African students who took part in

the CensusAtSchool survey project. The mean number of people living in a household was 6.208; the standard deviation was 2.576.

(a) Is the Normal condition met in this case? Justify your answer.

(b) Maurice claims that a 95% confidence interval for the population mean is $6.208\pm 1.96\frac{0.372}{\sqrt{47}}$. Explain why this interval is wrong. Then give the correct interval.

A milk processor monitors the number of bacteria per millilitre in raw milk received at the factory. A random sample of 10 one-millilitre specimens of milk supplied by one producer gives the following data:

5370 4890 5100 4500 5260 5150 4900 4760 4700 4870

Construct and interpret a 90% confidence interval for the population mean $\mu$.