7.7: Practice (Chapter 7)

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7.1: Point Estimates vs. Interval Estimates

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?
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 +- 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 x+-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%?
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?

7.2: Confidence Intervals for a Population Proportion

State the full formula for a confidence interval for a population proportion.
In a sample of 200 registered voters, 134 say they plan to vote in the next local election. Construct a 95% confidence interval for the proportion of all registered voters who plan to vote.
You survey 120 students and find that 45 of them get at least 8 hours of sleep regularly. Construct a 90% confidence interval for the population proportion.
Interpret the interval you just constructed — what does it say and what doesn't it say?
Why does it make sense that the standard error is smallest when p̂ is around 0.5?
Why is 0.5 used in place of p when determining the minimum sample size necessary for a proportion confidence interval?
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.
Can we construct a confidence interval if the sample is not random? Why or why not?
What role does independence play in calculating confidence intervals?
If np and n(1 − p) are both less than 10, what should you do?
Why should you be wary of surveys that do not report a margin of error?
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?
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. Calculate a 95% confidence interval for the proportion of Americans who access the internet on their cell phones

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?
What is the formula for the standard error of the sample mean?
When should you use the t-distribution instead of a normal (Z) distribution for means?
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. (Assume conditions are met.)
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?
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% con dent that the average waiting time of these 64 emergency room patients is between 128 and 147 minutes.
3. We are 95% con dent 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% con dent 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.
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 histogram below shows the distribution of the data from this sample. 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 4.72.44. Estimate the average age at first marriage of women using a 95% confidence interval, and interpret this interval in context. Discuss any relevant assumptions.

7.4: Effect of Sample Size and Confidence Level

What does the Z* value represent in a confidence interval formula?
What critical Z* values correspond to the following confidence levels?
- 90%
- 95%
- 99%
Explain how Z* relates to tail area (alpha) in confidence intervals.
How does increasing the confidence level from 90% to 99% affect your interval (even if sample data stays the same)?
Why do wider confidence levels (like 99%) result in a larger margin of error?
If you wanted a narrower interval, what could you do while keeping confidence level the same?
What is the effect of increasing the standard deviation on the margin of error?
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?
If a higher confidence level means that we are more confident about the number we are reporting, why don't we always report a confidence interval with the highest possible confidence level?
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?

7.5: Interpreting Confidence Intervals

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.
If a confidence interval for a population mean goes from 21.4 to 25.6, what is the point estimate?
What is the margin of error in the above interval?
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?
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.⁴³
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 introduced in Exercise 4.7 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?

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