math 105 test #4

type 2 error

a false negative and is made when the alternative hypothesis is true but the data does not provide convincing evidence that it is true

type 1 error

a false positive and is made when the null hypothesis is true but is rejected

Suppose a 95% confidence interval for the difference in test scores between Class 1 and Class 2 (in that order) is the following: 9 +/− 2. These results were based on independent samples of size 100 from each class. Now suppose you switch the order of Class 1 and Class 2 in your analysis but keep the data labeled correctly in terms of which class they came from. Which of the following statements is false? a) You are confident that the average for Class 2 is 7 to 11 points lower than for Class 1. b) You can't do it this way. You'll get negative numbers for the difference in the means and/or negative numbers for the standard error. c) You are still confident that the classes have significantly different mean scores. d) Your 95% confidence interval will now be entirely negative: from −7 to −11.

b

Suppose a confidence interval for the difference in mean weight loss for two different weight loss programs (Program 1 − Program 2) is entirely above zero. What does this mean? a) We can't say with any confidence that there is a difference in mean weight loss for the populations of people on these two programs. b) We can say with confidence that there is a difference in mean weight loss for the populations of people on these two programs; further, we can say that the average weight loss on Program 1 is higher. c) We can say with confidence that there is a difference in mean weight loss for the populations of people on these two programs; further, we can say that the average weight loss on Program 2 is higher. d) none of the above

b

Which of the following statements is false? a) Confidence intervals vary from one sample to the next. b) Confidence intervals are always close to their true population values. c) 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. d) None of the above statements are false.

b

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? a) 52% plus or minus 3% b) 52% plus or minus 2% c) 52% plus or minus 1% d) All of the above are "too close to call"

c

What is the most common level of confidence used to construct confidence intervals? a) 5% b) 90% c) 95% d) 100%

c

Which of the following are examples where you would be interested in estimating the population mean? a) What size was the viewing population that tuned in to the ABC News special last night? b) Do most people support or oppose the President's foreign policy? c) About how long do left-handed people live? d) all of the above

c

In Chapter 20, we saw that to construct a confidence interval for a population proportion it was enough to know the sample proportion and the sample size. Is the same true for constructing a confidence interval for a population mean? That is, is it enough to know the sample mean and sample size? Explain. a) Yes. You only need the mean and sample size. b) No. You also need the margin of error. c) No. You also need the sample proportion. d) No. You also need the sample standard deviation.

d

Which of the following statements is true regarding a 95% confidence interval? Assume numerous large samples are taken from the population. a) 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. b) 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. c) In 95% of all samples, the true proportion will fall within 2 standard deviations of the sample proportion. d) all of the above

d

______ sample sizes result in smaller standard errors and thus more narrow confidence intervals

larger

p-value

the probability that a standardized score would be as large as the computed test statistic value or even larger, if the null value is actually correct

rule for sample proportions

these sample proportions would have a bell-shaped frequency curve with the mean equal to the actual population proportion

If numerous large random samples or repetitions of the same size are taken from a population, the proportions from the various samples will have what approximate mean? a) 95% because most of them will be within 2 standard deviations of the true population value b) the true population average c) the true population proportion d) none of the above

c

hypothesis tests

can be used to decide whether sample data provide convincing evidence to reject a particular hypothesized value for a population proportion, mean, etc.

The _______ the desired confidence level, the _______ the multiplier and thus the wider the confidence interval.

higher, larger

when the alternative hypothesis is ______, a type 2 error could be made

true

when the null hypothesis is ______, a type 1 error could be made

true

If numerous large random samples or repetitions of the same size are taken from a population, the frequency curve made from proportions from the various samples will have what approximate shape? a) a bell shape b) unknown as it can change every time c) a bar graph with two bars, one for the proportion having the trait of interest and the other for the proportion not having the trait of interest d) a flat shape where each outcome should be equally likely

a

What numerical value do you calculate that gives you the answer to the question of how unlikely the test statistic would be if the null hypothesis were true? a) p-value b) confidence level c) Type II error d) Type I error

a

When someone reports that their results are found to be 'statistically significant', which type of statistical technique was most likely used? a) a hypothesis test b) a significance interval c) a confidence interval d) not enough information to tell

a

Explain whether you think the Rule for Sample Means applies to each of the following situations. If it does apply, specify the population of interest and the measurement of interest. If it does not apply, explain why not. (a) A university wants to know the average income of its alumni. Staff members select a random sample of 200 alumni and mail them a questionnaire. They follow up with a phone call to those who do not respond within 30 days. a) The rule applies. The population is alumni and the measurement is income. b) The rule applies. The population is staff members and the measurement is income. c) The rule does not apply because the measurement is not bell-shaped and the sample is not large enough. d) The rule does not apply because the sample is not random. (b) An automobile manufacturer wants to know the average price for which used cars of a particular model and year are selling in a certain state. They are able to obtain a list of buyers from the state motor vehicle division, from which they select a random sample of 20 buyers. They make every effort to find out what those people paid for the cars and are successful in doing so. a) The rule applies. The population is used cars of a particular model and year and the measurement is price. b) The rule applies. The population is used cars for sale and the measurement is number of miles on the car. c) The rule does not apply because the measurement is not bell-shaped and the sample is not large enough. d) The rule does not apply because the sample is not random.

a) a, b) c

Explain whether each of the following situations meets the conditions for which the Rule for Sample Proportions applies. If not, explain which condition is violated. (a)You are interested in knowing what proportion of days in typical years have rain or snow in the area where you live. For the months of January and February, you record whether there is rain or snow each day, and then you calculate the proportion of those days that had rain or snow. (Select all that apply.) a) This sample violates condition 1 because the outcome will not occur with a fixed relative-frequency probability. b) This sample violates condition 2 because the sample would not be random. c) This sample violates condition 3 because the size of the sample is not large enough. d) This sample meets all the conditions. (b) A large company wants to determine what proportion of its employees are interested in on-site day care. The company asks a random sample of 100 employees and calculates the sample proportion who are interested. (Select all that apply.) a) This sample violates condition 1 because there does not exist an actual population with a fixed proportion who have this certain opinion. b) This sample violates condition 2 because the sample would not be random. c) This sample violates condition 3 because the size of the sample is not large enough. d) This sample meets all the conditions.

a) a, b, b) d

Explain whether each of the following situations meets the conditions for which the Rule for Sample Proportions applies. If not, explain which condition is violated. (a) Unknown to the government, 10% of all cars in a certain city do not meet appropriate emissions standards. The government wants to estimate that percentage, so they take a random sample of 30 cars and compute the sample proportion that do not meet the standards. (Select all that apply.) a) This sample violates condition 1 because the outcome will not occur with a fixed relative-frequency probability. b) This sample violates condition 2 because the sample would not be random. c) This sample violates condition 3 because the size of the sample is not large enough. d) This sample meets all the conditions. (b) The Census Bureau would like to estimate what proportion of households have someone at home between 7 p.m. and 7:30 p.m. on weeknights, to determine whether that would be an efficient time to collect census data. The Bureau surveys a random sample of 2,000 households and visits them during that time to see whether someone is at home. (Select all that apply.) a) This sample violates condition 1 because the outcome will not occur with a fixed relative-frequency probability. b) This sample violates condition 2 because the sample would not be random. c) This sample violates condition 3 because the size of the sample is not large enough. d) This sample meets all the conditions.

a) c, b) d

When we revisited Case Study 6.4 in Chapter 21, we learned that a 95% confidence interval for the difference in years of education for mothers who did not smoke compared with those who did extended from 0.15 to 1.19 years, with higher education for those who did not smoke. Suppose we had used the data to construct a test instead of a confidence interval, to see if one group in the population was more educated than the other. What would the null and alternative hypotheses have been for the test? - null hypothesis a) Mothers who smoke have the same average level of education as those who do not smoke. b) Mothers who smoke have a higher average level of education as those who smoke. c) Mothers who smoke do not have the same average level of education as those who smoke. d) Mothers who smoke do not have a lower average level of education as those who do not smoke. e) Mothers who smoke always have a lower average level of education as those who smoke. - alternative hypothesis a) Mothers who smoke have the same average level of education as those who do not smoke. b) Mothers who smoke have a higher average level of education as those who smoke. c) Mothers who smoke do not have the same average level of education as those who smoke. d) Mothers who smoke do not have a lower average level of education as those who do not smoke. e) Mothers who smoke always have a lower average level of education as those who smoke.

a, c

Does working 5 hours a day or more at a computer contribute to deteriorating eyesight? - type 1 error a) Working 5 or more hours at a computer does not harm eyesight but the research concludes that it is harmful. Millions of workers would have needless concern. b) Working 5 or more hours at a computer does not harm eyesight and the research concludes that it is not harmful. Millions of workers would not be concerned. c) There really is a harmful effect and it is detected. Millions of workers who should be warned about this problem would learn about it. d) There really is a harmful effect but it goes undetected. Millions of workers who should be warned about this problem would not learn about it. - type 2 error a) Working 5 or more hours at a computer does not harm eyesight but the research concludes that it is harmful. Millions of workers would have needless concern. b) Working 5 or more hours at a computer does not harm eyesight and the research concludes that it is not harmful. Millions of workers would not be concerned. c) There really is a harmful effect and it is detected. Millions of workers who should be warned about this problem would learn about it. d) There really is a harmful effect but it goes undetected. Millions of workers who should be warned about this problem would not learn about it.

a, d

confidence interval

an interval of values used to estimate a population value

Given the convention of declaring that a result is statistically significant if the p-value is 0.05 or less, what decision would be made concerning the null and alternative hypotheses in each of the following cases? Be explicit about the wording of the decision. (a) p-value = 0.39 a) Reject the null hypothesis b) Do not reject the null hypothesis (b) p-value = 0.02 a) Reject the null hypothesis b) Do not reject the null hypothesis

b, a

In Case Study 1.2 and in Chapters 12 and 13, we examined a study showing that there appears to be a relationship between taking aspirin and incidence of heart attack. The null hypothesis in that study would be that there is no relationship between the two variables, and the alternative would be that there is a relationship. - Explain what a type 1 error would be for the study and what the consequence would be for the public. a) There is no relationship between aspirin and heart attack and the study does not find one. The use of aspirin would be unchanged. b) There is no relationship between aspirin and heart attack but the study finds one. People would start taking aspirin needlessly. c) Aspirin does help prevent heart attacks but the study does not discover the relationship, so people who could be helped by aspirin are not told to take it. d) Aspirin does help prevent heart attacks and the study discovers the relationship, so people who could be helped by aspirin are told to take it. - Explain what a type 2 error would be for the study and what the consequence would be for the public. a) There is no relationship between aspirin and heart attack and the study does not find one. The use of aspirin would be unchanged. b) There is no relationship between aspirin and heart attack but the study finds one. People would start taking aspirin needlessly. c) Aspirin does help prevent heart attacks but the study does not discover the relationship, so people who could be helped by aspirin are not told to take it. d) Aspirin does help prevent heart attacks and the study discovers the relationship, so people who could be helped by aspirin are told to take it.

b, c

In revisiting Case Study 5.4, we quoted the original journal article as reporting that "for any vertex baldness (i.e., mild, moderate, and severe combined), the age-adjusted RR was 1.4 (95% CI, 1.2 to 1.9)."† Interpret this result. Explain in words that someone with no training in statistics would understand. a) 95% of men with any vertex baldness are at a 1.2 to 1.9 times greater risk of heart attack than men with no baldness. b) 95% of men without baldness are at a 1.2 to 1.9 times greater risk of heart attack than men with any vertex baldness. c) With 95% confidence we can say that men with any vertex baldness are at a higher risk of heart attack than men with no baldness; their risk is 1.2 to 1.9 times greater. d) With 95% confidence we can say that men with any vertex baldness are at a higher risk of heart attack than men with no baldness; their risk is 1.4 times greater. e) With 95% confidence we can say that men without baldness are at a higher risk of heart attack than men with any vertex baldness; their risk is 1.2 to 1.9 times greater.

c

Suppose instead of comparing independent measurements taken from two groups, you used a matched-pairs experiment and one treatment is randomly assigned to each half of the pair. In this case, how should you compute the confidence interval for the difference? a) Do a separate confidence interval for each half of the pair and subtract the confidence intervals to get a range of differences. b) Do it the same way you would do it for two independent groups. You'll just get more accurate results because they came from matched pairs. c) Compute the differences for each pair, treat them as a single data set, and use the formula for a confidence interval for one mean (the mean difference). d) none of the above

c

Does placing babies in an incubator during infancy lead to claustrophobia in adult life? - type 1 error a) Incubators do cause a problem and it is detected. Those who were put in incubators would wonder why they had claustrophobia and would be told there was a reason, knowledge that might help them overcome it. b) Incubators do cause a problem but it goes undetected. Those who were put in incubators would wonder why they had claustrophobia and would not be told there was a reason, knowledge that might help them overcome it. c) Incubators do not cause claustrophobia but the research finds that they do. As a consequence, incubator use might be reduced or counseling offered to people who were really okay. d) Incubators do not cause claustrophobia and the research finds that they do not. As a consequence, incubator use might be reduced or counseling offered to people who were claustrophobic. - type 2 error a) Incubators do cause a problem and it is detected. Those who were put in incubators would wonder why they had claustrophobia and would be told there was a reason, knowledge that might help them overcome it. b) Incubators do cause a problem but it goes undetected. Those who were put in incubators would wonder why they had claustrophobia and would not be told there was a reason, knowledge that might help them overcome it. c) Incubators do not cause claustrophobia but the research finds that they do. As a consequence, incubator use might be reduced or counseling offered to people who were really okay. d) Incubators do not cause claustrophobia and the research finds that they do not. As a consequence, incubator use might be reduced or counseling offered to people who were claustrophobic.

c, b

Does placing plants in an office lead to fewer sick days? - type 1 error a) Plants do help reduce sick days, and the link is detected. A useful strategy would be used. b) Plants do help reduce sick days, but the link goes undetected. A useful strategy would go unused. c) Plants have no impact but the research finds that they do; as a consequence companies would needlessly offer plants to their employees. d) Plants have no impact and the research finds that they do not; as a consequence companies would not needlessly offer plants to their employees. - type 2 error a) Plants do help reduce sick days, and the link is detected. A useful strategy would be used. b) Plants do help reduce sick days, but the link goes undetected. A useful strategy would go unused. c) Plants have no impact but the research finds that they do; as a consequence companies would needlessly offer plants to their employees. d) Plants have no impact and the research finds that they do not; as a consequence companies would not needlessly offer plants to their employees.

c, b

In a report titled, "Secondhand Smoke: Is It a Hazard?", 26 studies linking secondhand smoke and lung cancer were summarized by noting, "those studies estimated that people breathing secondhand smoke were 8 to 150 percent more likely to get lung cancer sometime later."† Although it is not explicit, assume that the statement refers to a 95% confidence interval and interpret what this means. a) With 95% confidence we can say that people not breathing secondhand smoke are between 8% and 150% more likely to get lung cancer than those who do breathe it. b) 95% of people not breathing secondhand smoke are between 8% and 150% more likely to get lung cancer than those who do breathe it. c) 95% of people breathing secondhand smoke are between 8% and 150% more likely to get lung cancer than those who don't breathe it. d) With 95% confidence we can say that people breathing secondhand smoke are between 8% and 150% more likely to get lung cancer than those who don't breathe it. e) We do not have enough information to make a conclusion.

d

No matter what type of population value (or combination of population values) is being estimated using a confidence interval, what items should you be watching for in order to best assess the results? a) That the confidence level is clearly stated. b) That any real differences for which the researchers imply a causal relationship came from a randomized experiment. c) That the sample(s) selected represent the population(s). d) all of the above

d

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? a) A sample can be done more carefully than a complete audit. b) It is much cheaper. c) A well-designed sampling audit may yield a more accurate estimate than a less carefully carried out complete audit or census. d) all of the above

d

Suppose a 95% confidence interval for the difference in test scores between Class 1 and Class 2 (in that order) is the following: 9 +/− 2. These results were based on independent samples of size 100 from each class. What can you conclude? a) You are confident that the average for Class 1 was 7 to 11 points higher than for Class 2. b) You are confident that the averages for Class 1 and Class 2 are significantly different. c) You are confident that the observed difference found in the two samples (plus or minus the margin of error) will carry over to their respective populations. d) all of the above

d

Suppose a university wants to know the average income of its students who work, and all students supply that information when they register. Would the university need to use the methods in this chapter to compute a confidence interval for the population mean income? Explain. (Hint: What is the sample mean and what is the population mean?) a) Yes. The confidence interval will allow the university to estimate student income with a measure of certainty. b) Yes. It would be too time-consuming to compute the average for the entire population. c) No. The university cannot assume that student income is independent and must use other methods. d) No. They would have the necessary information on the entire population so they could compute it directly.

d

Suppose you computed a 95% confidence interval for the difference in mean weight between two species of snakes in a large nature reserve (species #1 − species #2), and your interval is −3.6 to 61.6 ounces. What can you conclude? a) If we were willing to use 90% confidence, we could say that the observed difference in the sample means represents a real difference in the population means. b) You cannot say, even with 95% confidence, that the observed difference in sample means represents a real difference in the population means. c) Because the interval extends so much further in the positive direction than the negative, the evidence suggests that species #1 weighs more than species #2 on average, but we can't say for sure. d) all of the above

d

When a relationship or value from a sample is so strong that we decide to rule out chance as an explanation for its magnitude, what does this mean? a) We conclude that the observed result carries over to the population and cannot be explained away by chance. b) We could have been unlucky with our sample, and come to the wrong conclusion, but that chance is small. c) The observed result is statistically significant. d) all of the above

d

Which of the following describes the power of a test? a) It is higher when the population value is farther from the value in the null hypothesis. b) Bigger samples result in more power for the test, and smaller samples result in less power for the test. c) It is the probability of making the correct decision when the alternative hypothesis is true. d) all of the above

d

Which of the following is true when a Type I error has been committed? a) The data must have convinced us that the alternative hypothesis was true. b) The probability of making a Type I error is equal to the stated level of significance, usually 0.05. c) The null hypothesis has to have been true. d) all of the above

d