CHAPTER 20 Name:

estimating proportions with confidence

  1. Explain (in words that a non-statistics student would understand) what is meant by a ‘95% confidence interval.’
  1. Statisticians have a phrase, “Being a statistician means never having to say you’re certain.” Do you know whether the confidence interval constructed by your sample actually contains the true population value? Why or why not?
  1. A 95% confidence interval means that 95% of all possible random samples will result in an interval that contains the true population value, and 5% of them won’t. How could a confidence interval that is based on a random sample not contain the true population value?
  1. What is the most common level of confidence used to construct confidence intervals?
  2. 5%
  3. 90%
  4. 95%
  5. 100%
  1. Which of the following is a correct interpretation of a 90% confidence interval?
  2. 90% of the random samples you could select would result in intervals that contain the true population value.
  3. 90% of the population values should be close to our sample results.
  4. Once a specific sample has been selected, the probability that its resulting confidence interval contains the true population value is 90%.
  5. All of the above statements are true.
  1. Which of the following statements is false?
  2. Confidence intervals are always close to their true population values.
  3. Confidence intervals vary from one sample to the next.
  4. The key to constructing confidence intervals is to understand what kind of dissimilarity we should expect to see in various samples from the same population.
  5. None of the above statements are false.
  1. How does a 90% confidence interval compare to a 95% confidence interval?
  2. Fewer of the samples will result in intervals that contain the true population value in the 90% case.
  3. Fewer of the samples will result in incorrect intervals in the 90% case.
  4. With the 90% confidence interval you are less willing to take a chance on missing the true value.
  5. All of the above.
  1. A(n) ______is an interval of values computed from the sample data that is almost sure to cover the true population number.
  1. Give an example of a confidence interval for a proportion with a margin of error of 5% that falls below zero for some of its values.
  1. Name two different situations that can result in a confidence interval for a proportion falling into negative numbers for some of its values (which does not make sense). Give an example of each.
  1. Which would be wider, a 90% confidence interval or a 95% confidence interval? (Assume both of them were calculated using the same sample data.) Explain your answer.
  1. Suppose a survey was conducted to find out what proportion of Americans intend to vote in the next Presidential election. For which of the following confidence intervals would it be fair to conclude, with high confidence, that a majority of Americans will vote in the next Presidential election?
  2. 52% plus or minus 3%
  3. 52% plus or minus 2%
  4. 52% plus or minus 1%
  5. All of the above are “too close to call.”

Narrative: Bread machines

Suppose a survey was done this year to find out what percentage of all Americans own a bread machine. Out of their random sample of 1,000 Americans, 317 own a bread machine. The margin of error for this survey was plus or minus 3%.

  1. {Bread machine narrative} What proportion of the entire American population owns a bread machine, based on these results?
  2. Definitely 317/1,000 = .317 or 31.7%
  3. Probably between 28.7% and 34.7%
  4. 31.7% times the population of the U.S.
  5. None of the above.
  1. {Bread machine narrative} Suppose three years ago, 29% of Americans owned a bread machine. Based on the results of this current survey, what would you conclude (with high confidence) about the population of all Americans now compared with three years ago?
  2. A larger proportion of Americans own a bread machine now compared to three years ago.
  3. A smaller proportion of Americans own a bread machine now compared to three years ago.
  4. The proportion of Americans who own a bread machine has not significantly changed from three years ago.
  5. None of the above.
  1. Suppose the U.S. government reports that the total number of people in the U.S. who are currently infected with HIV is likely to be between 300,000 and 1,000,000. What is the margin of error for these findings? (Assume a symmetric 95% confidence interval.)
  2. +/ 350,000
  3. +/ 700,000
  4. +/ 5%
  5. Not enough information to tell

Narrative: Oranges

Suppose a shipment of oranges is advertised to weigh 5 pounds per bag. We know that not every bag can contain exactly 5 pounds of oranges. We decide to take a random sample of 100 bags of oranges and find out what they tell us about the population of all bags in this shipment. We are only interested in whether or not the bags are underweight, so each bag is weighed and counted as underweight if it weighs less than 5 pounds. Five bags in our sample of 100 were found to be underweight.

  1. {Oranges narrative} What percentage of all the bags in the entire shipment do you think are underweight? Give the most complete answer you can.
  1. {Oranges narrative} What is the margin of error for a 95% confidence interval for the proportion of bags in the shipment that are underweight?
  1. {Oranges narrative} Suppose the grocery store who ordered the oranges will reject the shipment if they believe, based on these sample results, that more than 10% of the bags in the entire truckload are underweight. Based on our sample, will they have to return this shipment? Explain your answer.
  1. Using one divided by the square root of the sample size is known as a ‘conservative’ formula for the margin of error for a sample proportion. Explain what that means.
  1. The formula for calculating a confidence interval for a population proportion is based on the rule of sample proportions, which has assumptions that need to be met. What is the most important assumption that you need to check before applying the confidence interval formula to a sample proportion?
  1. Which of the following statements is true regarding a 95% confidence interval? Assume numerous large samples are taken from the population.
  2. In 95% of all samples, the sample proportion will fall within 2 standard deviations of the mean, which is the true proportion for the population.
  3. In 95% of all samples, the true proportion will fall within 2 standard deviations of the sample proportion.
  4. If we add and subtract 2 standard deviations to/from the sample proportion, in 95% of all cases we will have captured the true population proportion.
  5. All of the above.
  1. Sampling methods and confidence intervals are routinely used for financial audits of large companies. Which of the following is an advantage of doing it this way, versus having a complete audit of all records?
  2. It is much cheaper.
  3. A sample can be done more carefully than a complete audit.
  4. A well-designed sampling audit may yield a more accurate estimate than a less carefully carried out complete audit or census.
  5. All of the above.
  1. In which of the following situations can you construct a confidence interval for the population proportion with only what is given?
  2. The sample proportion and the margin of error.
  3. The sample proportion and the sample size.
  4. The number of individuals in the sample with the trait of interest, and the total sample size.
  5. All of the above.
  1. In which of the following situations can a confidence interval provide useful information for making a decision?
  2. To assess whether or not at least 90% of a company’s financial entries are correct.
  3. To estimate the percentage of married couples in which the wife is taller than the husband.
  4. To help determine whether or not someone has ESP (versus just being a lucky guesser).
  5. All of the above.
  1. To obtain a 99.7% (virtually all-encompassing) confidence interval for the true population proportion, you would add and subtract about ______standard deviations to/from the sample proportion.
  1. Using the same sample data, the margin of error for an 80% confidence interval is ______than the margin of error for a 90% confidence interval.