7.8: Practice (Chapter 7)

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7.1: Point Estimates vs. Interval Estimates

A single-value estimate for a population parameter is called a:
1. Confidence Interval
2. Standard Deviation
3. Point Estimate
4. Population Mean
A 2023 Gallup poll of 1000 Americans found that 45% answered "Three or more" to the question "What is the ideal number of children to have?" The margin of error of the poll was 3%.
1. What is the sample statistic?
2. What is the population parameter we wish to estimate?
True or False: A point estimate is a single numerical approximation for a population parameter while a confidence interval is a range of values with a level of confidence.
Determine if the following are point estimates or interval estimates?
1. The 2010 General Social Survey asked 1,259 US residents: "Do you think the use of marijuana should be made legal, or not?" 48% of the respondents said it should be made legal.
2. The true proportion of people whose favorite color is black is between 2% and 6%.
3. The average weight of a bag of chips is 1.2 oz $ \pm $ 0.1oz.
4. The proportion of those who watch The Daily Show is (0.07, 0.15).
5. Daily consumer spending for the six-day period after Thanksgiving averaged $84.71.
Express the interval (339.8, 548.6) in the form $ \bar{x} \pm ME $
Express the interval (59%, 67%) in the form $ \hat{p} \pm ME $
Express the interval estimate $ 6.379 \pm 6.86 $ in interval notation
Express the interval $ 595.4 < \mu < 851.8 $ in the form $ \bar{x} \pm ME $
What is the interval estimate for a poll where 45% answered "yes" to a question if the margin of error of the poll was 3%?
Express the interval estimate $ 0.35 \pm 0.05 $ in interval notation
If a confidence interval for a population mean goes from 21.4 to 25.6, what is the point estimate?
For a class project, a political science student at a large university wants to estimate the percentage of students who are registered voters. He surveys 279 students and finds that 54% are registered voters. Answer the following questions.
1. What is the population proportion?
2. What is the point estimate for the population proportion?
3. How many students are registered voters?
4. What is the sample size?
In a random sample of 419 patients in a hospital emergency room, the mean wait time was 42.3 minutes with a standard deviation of 7 minutes. Answer the following questions.
1. What is the population mean wait time for all emergency room patients at this hospital?
2. What is the point estimate for the population mean?
3. What is the population standard deviation?
4. What is the sample standard deviation?
How do you differentiate between a point estimate and an interval estimate?
How do you find the margin of error from the lower and upper bounds of a confidence interval?

7.2: Confidence Intervals for a Population Proportion

True or False: A confidence interval is used to estimate a parameter using data collected from a sample.
State the full formula for a confidence interval for a population proportion.
Can we construct a confidence interval if the sample is not random? Why or why not?
Name the three conditions that should be met before constructing a confidence interval for a proportion.
A polling agency finds that 480 out of 1,200 respondents favor a new policy. Check whether it's appropriate to use a normal model before constructing a confidence interval.
If $n \hat{p} $ and $ n (1 − \hat{p}) $ are both less than 10, what should you do?
Find the critical value for the common confidence levels of 90%, 95%, and 99%.
Describe the critical value and margin of error in your own words.
What does the critical value represent in a confidence interval formula?
What is the relationship between the standard error and margin of error?
Why should you be wary of surveys that do not report a margin of error?
In a sample of 200 registered voters, 134 say they plan to vote in the next local election.
1. What is the point estimate for the population proportion?
2. Verify that the conditions are met.
3. What is the critical value for a 95% confidence level?
4. What is the margin of error?
5. What is the 95% confidence interval for the proportion of all registered voters who plan to vote?
6. State the conclusion in a complete sentence.
A 2012 survey of 2,254 American adults indicates that 17% of cell phone owners do their browsing on their phone rather than a computer or other device.
1. What is the point estimate for the population proportion?
2. Verify that the conditions are met.
3. What is the critical value for a 90% confidence level?
4. What is the margin of error?
5. What is the 90% confidence interval for the population proportion?
6. State the conclusion in a complete sentence.
You survey 120 students and find that 45 of them get at least 8 hours of sleep regularly.
1. What is the point estimate for the population proportion?
2. Verify that the conditions are met.
3. What is the critical value for a 99% confidence level?
4. What is the margin of error?
5. What is the 99% confidence interval for the population proportion?
6. State the conclusion in a complete sentence
Suppose that an insurance company conducted a survey. They randomly surveyed 400 drivers and found that 320 claimed they always buckle up. We are interested in the population proportion of drivers who claim they always buckle up.
1. Which distribution should you use for this problem? Explain your choice.
2. Construct a 95% confidence interval for the population proportion who claim they always buckle up
  1. Calculate the error bound.
  2. Sketch a graph of the interval estimate
  3. State the confidence interval.
3. If this survey were done by telephone, list three difficulties we might have in obtaining random results.
According to a recent survey of 1,200 people, 16% feel that the president is doing an acceptable job. We are interested in the population proportion of people who feel the president is doing an acceptable job.
1. Which distribution should you use for this problem? Explain your choice.
2. Construct a 90% confidence interval for the population proportion who claim they always buckle up
  1. Calculate the error bound.
  2. Sketch a graph of the interval estimate
  3. State the confidence interval.
3. If this survey were done by telephone, list three difficulties we might have in obtaining random results.
A telephone poll of 1,000 adult Americans was reported in an issue of Time Magazine. One of the questions asked was “What is the main problem facing the country?” Twenty percent answered “crime.” We are interested in the population proportion of adult Americans who feel that crime is the main problem.
1. Construct a 95% confidence interval for the population proportion of adult Americans who feel that crime is the main problem.
2. Suppose we want to lower the sampling error. What is one way to accomplish that?
3. The sampling error given by Yankelovich Partners, Inc. (which conducted the poll) is ±3%. In one to three complete sentences, explain what the ±3% represents.
In a 2010 Survey USA poll, 70% of the 119 respondents between the ages of 18 and 34 said they would vote in the 2010 general election for Prop 19, which would change California law to legalize marijuana and allow it to be regulated and taxed. At a 95% confidence level, this sample has an 8% margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning.
1. We are 95% confident that between 62% and 78% of the California voters in this sample support Prop 19.
2. We are 95% confident that between 62% and 78% of all California voters between the ages of 18 and 34 support Prop 19.
3. If we considered many random samples of 119 California voters between the ages of 18 and 34, and we calculated 95% confidence intervals for each, 95% of them will include the true population proportion of Californians who support Prop 19.
4. In order to decrease the margin of error to 4%, we would need to quadruple (multiply by 4) the sample size.
5. Based on this confidence interval, there is sufficient evidence to conclude that a majority of California voters between the ages of 18 and 34 support Prop 19.
On June 28, 2012 the U.S. Supreme Court upheld the much debated 2010 healthcare law, declaring it constitutional. A Gallup poll released the day after this decision indicates that 46% of 1,012 Americans agree with this decision. At a 95% confidence level, this sample has a 3% margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning.
1. We are 95% confident that between 43% and 49% of Americans in this sample support the decision of the U.S. Supreme Court on the 2010 healthcare law.
2. We are 95% confident that between 43% and 49% of Americans support the decision of the U.S. Supreme Court on the 2010 healthcare law.
3. If we considered many random samples of 1,012 Americans, and we calculated the sample proportions of those who support the decision of the U.S. Supreme Court, 95% of those sample proportions will be between 43% and 49%.
A Washington Post article from 2009 reported that "support for a government-run health-care plan to compete with private insurers has rebounded from its summertime lows and wins clear majority support from the public." More specifically, the article says "seven in 10 Democrats back the plan, while almost nine in 10 Republicans oppose it. Independents divide 52 percent against, 42 percent in favor of the legislation." There were were 819 Democrats, 566 Republicans and 783 Independents surveyed.
1. Create a 90% confidence interval for the proportion of Independents who oppose the public option plan.
2. A political pundit on TV claims that a majority of Independents oppose the health care public option plan. Do these data provide strong evidence to support this statement?

7.3: Confidence Intervals for a Population Mean

What changes in the construction of a confidence interval when estimating a mean instead of a proportion?
A family of distributions resembling the normal distribution but with thicker tails best describes
1. the bootstrap
2. degrees of freedom
3. t-distribution
4. confidence intervals
When should you use the t-distribution instead of a normal (Z) distribution for means?
Explain how the shape of the t-distribution affects your margin of error compared to a Z-based interval.
Why would you be more likely to use a T-interval in a real-world situation than a Z-interval?
What is the formula for the standard error of the sample mean?
An e-business manager wants to test the claim that the mean exposure to a certain type of internet ads exceeds 1000 viewers per day. Randomly selecting 20 days she calculated the sample mean and standard deviation.
1. If the population standard deviation is not available, which distribution should she use?
2. What is the degrees of freedom in this situation?
Given degree of freedom df = 16, find the probability by using the Student's t distribution: $ P(t>0.4) = $
You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 54. Find the critical value that corresponds to a confidence level of 99.5%.
You intend to estimate a population mean with a confidence interval. You believe the population to have a normal distribution. Your sample size is 24. While it is an uncommon confidence level, find the critical value that corresponds to a confidence level of 93%.
The average time students spend on homework per week is estimated from a sample of 15 students, who report an average of 13.2 hours with a standard deviation of 4.8 hours. Construct a 95% confidence interval.
A survey was conducted on 203 undergraduates from Duke University who took an introductory statistics course in Spring 2012. Among many other questions, this survey asked them about the number of exclusive relationships they have been in. The sample average is 3.2 with a standard deviation of 1.97. Estimate the average number of exclusive relationships Duke students have been in using a 90% confidence interval and interpret this interval in context. Check any conditions required for inference, and note any assumptions you must make as you proceed with your calculations and conclusions.
The National Survey of Family Growth conducted by the Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy, infertility, use of contraception, and men's and women's health. One of the variables collected on this survey is the age at first marriage. The histogram below shows the distribution of ages at first marriage of 5,534 randomly sampled women between 2006 and 2010. The average age at first marriage among these women is 23.44 with a standard deviation of 4.72. Estimate the average age at first marriage of women using a 95% confidence interval, and interpret this interval in context. Discuss any relevant assumptions.
Karen wants to advertise how many chocolate chips are in each Big Chip cookie at her bakery. She randomly selects a sample of 42 cookies and finds that the number of chocolate chips per cookie in the sample has a mean of 7.8 and a standard deviation of 2.3. What is the 98% confidence interval for the number of chocolate chips per cookie for Big Chip cookies?
Kyla wants to estimate the average number of hours worked per shift by ER nurses across a large hospital network. The population is known to be approximately normally distributed. Kyla takes a random sample of 19 nurse ER shifts and finds that the sample average is 10.5 hours, with a standard deviation of 0.6 hours.
The effectiveness of a blood-pressure drug is being investigated. An experimenter finds that, on average, the reduction in systolic blood pressure is 42.3 for a sample of size 434 and standard deviation 16.6. Estimate how much the drug will lower a typical patient's systolic blood pressure (using a 98% confidence level).

You must estimate the mean temperature (in degrees Fahrenheit) with the following sample temperatures. Find the 98% confidence interval.

57.4

86.3

51.9

51.6

99.5

86.5

33.9

84.9

69.2

72.6

67.9

41.9

You measure 21 randomly selected textbooks' weights, and find they have a mean weight of 35 ounces. Assume the population standard deviation is 14.5 ounces. Based on this, construct a 95% confidence interval for the true population mean textbook weight.
A veterinarian wants to estimate the mean weight of a cat in their city. They take a random sample of 16 cats and found a mean of 9.6 lbs with a sample standard deviation of 0.8 lbs. Find the 95% confidence interval of the mean. Assume that the cat weights are normally distributed.

A sample of the length in inches for newborns is given below. Assume that lengths are normally distributed. Find the 90% confidence interval of the mean length.

15.2	22.3	22.5	19.4	18.9	17.7
16.4	23	22.2	21

Private colleges and universities rely on money contributed by individuals and corporations for their operating expenses. Much of this money is invested in a fund called an endowment, and the college spends only the interest earned by the fund. A recent survey of 43 private colleges in the United States revealed the following endowments (in millions of dollars):

Data
276.8	410.9	313.8	243.3	218	87.5	234.5	431	236.8	71.3
151.5	65.6	370.1	88.2	325.8	69.2	97.7	179.5	91.9	180.2
99.8	129.7	257	176.5	174.2	219.2	241	213.4	188.1	64
226.2	34	6.3	253	334.9	301.5	203.1	21	260.1	235.1
194.1	18.6	90.2

Summary statistics yield the sample mean 188.01 millions of dollars and the sample standard deviation 107.07 millions of dollars. Construct and interpret a 95% confidence interval for the mean endowment of all private colleges in the United States.

Suppose you use simple random sampling to select and measure 46 backpacks' weights, and find they have a mean weight of 45 ounces. Assume the population standard deviation is 8 ounces. Based on this, construct a 90% confidence interval for the true population mean backpack weight.

7.4: Effect of Sample Size and Confidence Level

How does increasing the confidence level affect a confidence interval?
Why do higher confidence levels result in a larger margin of error?
What is the effect of decreasing the sample size on confidence interval?
Why do larger sample sizes make the margin of error smaller?
What is the effect of increasing the standard deviation on the margin of error?
Give an example of a situation where you would use each of the following confidence levels: 90%, 95%, and 99%
How does a statistician decide on which confidence level to use?
Why would we not use a 99.99% confidence intervals?
Suppose we calculated the 99% confidence interval for the average age of runners in the 2012 Cherry Blossom Run as (32.7, 37.4) based on a sample of 100 runners. How could we decrease the width of this interval without losing confidence?
We are interested in estimating the proportion of graduates at a mid-sized university who found a job within one year of completing their undergraduate degree. Suppose we conduct a survey and nd out that 348 of the 400 randomly sampled graduates found jobs. The graduating class under consideration included over 4500 students.
1. Describe the population parameter of interest. What is the value of the point estimate of this parameter?
2. Check if the conditions for constructing a confidence interval based on these data are met.
3. Calculate a 95% confidence interval for the proportion of graduates who found a job within one year of completing their undergraduate degree at this university, and interpret it in the context of the data.
4. What does "95% confidence" mean?
5. Now calculate a 99% confidence interval for the same parameter and interpret it in the context of the data.
6. Compare the widths of the 95% and 99% confidence intervals. Which one is wider? Explain.
Greece has faced a severe economic crisis since the end of 2009. A Gallup poll surveyed 1,000 randomly sampled Greeks in 2011 and found that 25% of them said they would rate their lives poorly enough to be considered "suffering".
1. Describe the population parameter of interest. What is the value of the point estimate of this parameter?
2. Check if the conditions required for constructing a confidence interval based on these data are met.
3. Construct a 95% confidence interval for the proportion of Greeks who are "suffering".
4. Without doing any calculations, describe what would happen to the confidence interval if we decided to use a higher confidence level.
5. Without doing any calculations, describe what would happen to the confidence interval if we used a larger sample.
In 2010 the General Social Survey asked 1,259 US residents and reported a sample where about 48% of US residents thought marijuana should be made legal. If we wanted to limit the margin of error of a 95% confidence interval to 2%, about how many Americans would we need to survey ?
A poll of 783 Independents surveyed evaluating support for the health care public option in 2009, reporting that 52% of Independents in the sample opposed the public option. If we wanted to estimate this number to within 1% with 90% confidence, what would be an appropriate sample size?
SAT scores are distributed with a mean of 1,500 and a standard deviation of 300. You are interested in estimating the average SAT score of first year students at your college. If you would like to limit the margin of error of your 95% confidence interval to 25 points, how many students should you sample?
A political candidate has asked you to conduct a poll to determine what percentage of people support her. If the candidate only wants a 3% margin of error at a 95% confidence level, what size of sample is needed?
You want to obtain a sample to estimate the proportion of a population that possess a particular genetic marker. Based on previous evidence, you believe approximately 44% of the population have the genetic marker. You would like to be 98% confident that your estimate is within 3% of the true population proportion. How large of a sample size is required?
Insurance companies are interested in knowing the population percent of drivers who always buckle up before riding in a car.
1. When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 95% confident that the population proportion is estimated to within 0.03?
2. If it were later determined that it was important to be more than 95% confident and a new survey was commissioned, how would that affect the minimum number you would need to survey? Why?
Why is 0.5 used in place of $ \hat{p} $ when determining the minimum sample size necessary for a proportion confidence interval?

7.5: Interpreting Confidence Intervals

True or False: If one million 95% confidence intervals for the mean were computed, then approximately 95% of these confidence intervals would contain the population mean.
Describe the meaning of a 95% confidence level in the context of estimating a population proportion.
Which of the following is the best interpretation of a 95% confidence interval?
1. There's a 95% chance that the true population parameter is within this one specific interval.
2. 95% of all possible confidence intervals constructed from samples like this one will contain the true parameter.
3. This sample is 95% accurate.
4. We are 95% sure our sample statistic is correct.
Which of the following would be the best interpretation of the 99% confidence level?
1. If we were to take 100 samples, then approximately 95 would contain the true value of the mean and approximately 5 would not
2. If we were to take 100 samples, then approximately 99 would contain the true value of the mean and approximately 1 would not.
3. If we were to take 100 samples, then exactly 99 would contain the true value of the mean and exactly 1 would not
4. There is a 99% probability that the confidence interval contains the true value of the mean
Explain why we do not say “there is a 95% probability the population mean falls in this interval.”
Why is it misleading to say that a 100% confidence interval would definitely include the population parameter?
A hospital administrator hoping to improve wait times decides to estimate the average emergency room waiting time at her hospital. He collects a simple random sample of 64 patients and determines the time (in minutes) between when they checked in to the ER until they were first seen by a doctor. A 95% confidence interval based on this sample is (128 minutes, 147 minutes), which is based on the normal model for the mean. Determine whether the following statements are true or false, and explain your reasoning for those statements you identify as false.
1. This confidence interval is not valid since we do not know if the population distribution of the ER wait times is nearly normal.
2. We are 95% confident that the average waiting time of these 64 emergency room patients is between 128 and 147 minutes.
3. We are 95% confident that the average waiting time of all patients at this hospital's emergency room is between 128 and 147 minutes.
4. 95% of such random samples would have a sample mean between 128 and 147 minutes.
5. A 99% confidence interval would be narrower than the 95% confidence interval since we need to be more sure of our estimate.
6. The margin of error is 9.5 and the sample mean is 137.5.
7. In order to decrease the margin of error of a 95% confidence interval to half of what it is now, we would need to double the sample size.
The 2009 holiday retail season, which kicked off on November 27, 2009 (the day after Thanksgiving), had been marked by somewhat lower self-reported consumer spending than was seen during the comparable period in 2008. To get an estimate of consumer spending, 436 randomly sampled American adults were surveyed. Daily consumer spending for the six-day period after Thanksgiving, spanning the Black Friday weekend and Cyber Monday, averaged $84.71. A 95% confidence interval based on this sample is ($80.31, $89.11). Determine whether the following statements are true or false, and explain your reasoning.
1. We are 95% con dent that the average spending of these 436 American adults is between $80.31 and $89.11.
2. This confidence interval is not valid since the distribution of spending in the sample is right skewed.
3. 95% of such random samples would have a sample mean between $80.31 and $89.11.
4. We are 95% confident that the average spending of all American adults is between $80.31 and $89.11.
5. A 90% confidence interval would be narrower than the 95% confidence interval since we don't need to be as sure about capturing the parameter.
6. In order to decrease the margin of error of a 95% confidence interval to a third of what it is now, we would need to use a sample 3 times larger.
7. The margin of error for the reported interval is 4.4.
The General Social Survey (GSS) is a sociological survey used to collect data on demographic characteristics and attitudes of residents of the United States. In 2010, the survey collected responses from 1,154 US residents. The survey is conducted face-to-face with an in-person interview of a randomly-selected sample of adults. One of the questions on the survey is "After an average work day, about how many hours do you have to relax or pursue activities that you enjoy?" A 95% confidence interval from the 2010 GSS survey is 3.53 to 3.83 hours.
1. Interpret this interval in the context of the data.
2. What does a 95% confidence level mean in this context?
3. Suppose the researchers think a 90% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval? Assume the standard deviation has remained constant since 2010.
Another question on the General Social Survey is "For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010.
1. Interpret this interval in context of the data.
2. What does a 95% confidence level mean in this context?
3. Suppose the researchers think a 99% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval?
4. If a new survey asking the same questions was to be done with 500 Americans, would the standard error of the estimate be larger, smaller, or about the same. Assume the standard deviation has remained constant since 2010.
The Marist Poll published a report stating that 66% of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age. It was also reported that interviews were conducted on 1,018 American adults, and that the margin of error was 3% using a 95% confidence level.
1. Verify the margin of error reported by The Marist Poll.
2. Based on a 95% confidence interval, does the poll provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65?
A survey on 1,509 high school seniors who took the SAT and who completed an optional web survey between April 25 and April 30, 2007 shows that 55% of high school seniors are fairly certain that they will participate in a study abroad program in college.⁴³
1. Is this sample a representative sample from the population of all high school seniors in the US? Explain your reasoning.
2. Let's suppose the conditions for inference are met. Even if your answer to part (a) indicated that this approach would not be reliable, this analysis may still be interesting to carry out (though not report). Construct a 90% confidence interval for the proportion of high school seniors (of those who took the SAT) who are fairly certain they will participate in a study abroad program in college, and interpret this interval in context.
3. What does "90% confidence" mean?
4. Based on this interval, would it be appropriate to claim that the majority of high school seniors are fairly certain that they will participate in a study abroad program in college?
In a recent Zogby International Poll, 5 of 48 respondents rated the likelihood of a terrorist attack in their community as “likely” or “very likely.” Create a 97% confidence interval for the proportion of American adults who believe that a terrorist attack in their community is likely or very likely. Explain what this confidence interval means in the context of the problem.
In a survey, 600 adults in a certain country were asked how many hours they worked in the previous week. Based on the results, a 95% confidence interval for mean number of hours worked was lower bound: 34 hours and upper bound: 39 hours. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw.
1. There is a 95% chance the mean number of hours worked by adults in this country in the previous week was between 34 hours and 39 hours.
2. We are 95% confident that the mean number of hours worked by adults in a particular area of this country in the previous week was between 34 hours and 39 hours.
3. 95% of adults in this country worked between 34 hours and 39 hours last week.
4. We are 95% confident that the mean number of hours worked by adults in this country in the previous week was between 34 hours and 39 hours.
A recent survey estimated that 19 percent of all people living in a certain region regularly use sunscreen when going outdoors. The margin of error for the estimate was 1 percentage point. Based on the estimate and the margin of error, which of the following is an appropriate conclusion?
1. Approximately 1% of all the people living in the region were surveyed.
2. Between 18% and 20% of all the people living in the region were surveyed.
3. All possible samples of the same size will result in between 18% and 20% of those surveyed indicating they regularly use sunscreen.
4. The probability is 0.01 that a person living in the region will use sunscreen when going outdoors.
5. It is plausible that the percent of all people living in the region who regularly use sunscreen is 18.5%.
An advertiser claims that 9 out of 10 dentists (90%) recommend Brand X toothpaste. A consumer group investigates the truth of this claim. They survey a randomly selected sample of 4500 dentists, and find that 3964 of them recommend Brand X toothpaste. Based on this survey, they construct a 95% confidence interval for the proportion of dentists that recommend Brand X toothpaste of (0.871, 0.890). What can be concluded from this survey?

Practice problems include parts of:

Introductory Statistics 2e by OpenStax is licensed under CC BY 4.0

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

and

OpenIntro Statistics by David Diez, Christopher Barr, & Mine Çetinkaya-Rundel is licensed under CC BY 3.0

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