Chapter 21 Solutions and Mini-Project Notes

Chapter 21 Solutions and Mini-Project Notes

CHAPTER 21 SOLUTIONS AND MINI-PROJECT NOTES

CHAPTER 21

THE ROLE OF CONFIDENCE INTERVALS IN RESEARCH

EXERCISE SOLUTIONS

21.1No. You also need the sample standard deviation.

21.2Using Example 1, the sample mean weight loss for the 42 men who dieted was 7.2 kg. The corresponding population mean is the hypothetical average weight loss that would be achieved if all men similar to this group were to follow a diet similar to the one used by this group, for the same amount of time.

21.3a.The standard error of the mean is 0.5/ = 0.5/6 = 1/12 or 0.083 hours or 5 minutes.

b.A 95% confidence interval is 12 hrs 50 min to 13 hrs 10 min.

c.The SEM for the Dutch babies is 0.062, so the "measure of variability" is the square root of (0.083)2 + (0.062)2 = 0.104. A 95% confidence interval for the difference is 2 ± 2(0.104) or 1.79 to 2.21 hours.

21.45%.

21.5No. They would have the necessary information on the entire population so they could compute it directly.

21.6a.We could be 95% confident that the two population means are not equal.

b.The population means could be (but are not necessarily) the same.

21.7a.With 95% confidence we could conclude that the risk of disease was greater under one of the conditions than the other.

b.We could not conclude that the risks under the two conditions were different (but we would not go so far as to say they are equal.)

21.8a.A 90% confidence interval is 29 ± (1.645)(16.3) or 2.2 to 55.8.

b.Yes, because this interval does not cover zero. (The 95% confidence interval did cover 0.)

21.9a.A 95% confidence interval extends from 64 to 112.

b.A 95% confidence interval extends from 95 to 139.

c.No. Because there is overlap in the intervals we cannot be confident that the true means are indeed different.

21.10This would have been misleading because the interval does not lie entirely above zero, so there is a good chance that the difference in the sample means does not represent a real difference in the population means at all, or that there is a difference in the opposite direction.

21.11a.The confidence interval is 64.64 ± 0.81 or 63.83 to 65.45.

b.The confidence interval is 63.97 ± 2.0 or 61.97 to 65.97.

c.The interval in part b is wider because the sample size is smaller. We could not conclude that there is a difference in the mean ages because there is overlap in the two intervals.

d.The interval is (64.64 − 63.97) ± 2(1.08) or −1.49 to 2.83. This interval does not lie entirely above zero, so there is some chance that the difference in these sample means does not represent a real difference in the population of strong left-handers and right-handers.

21.12The statement means that we know, with 95% certainty, 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.

21.13The study says that with 95% certainty we know that people breathing secondhand smoke are between 8% and 150% more likely to get lung cancer than those who don't breathe it.

21.14This study exhibited a large placebo effect, meaning that scores changed even for those women taking a placebo instead of calcium. A comparison of the calcium group third cycle scores to their own baseline scores would include both the placebo effect and the actual impact of taking calcium. Thus, it would overrate the influence of taking calcium.

21.15a.Mood swings: .07 to .33

b.Crying spells: .05 to .23

c.Aches and pains: .08 to .28

d.Craving sweets or salts: .08 to .20

21.16The confidence interval is 12% to 22%. This interval lies entirely above zero, so it does indicate that opinion on this issue among American adults had changed.

21.17Method 1; the difference within couples is desired, not across sexes. We can be 95% certain that on average British husbands are between 1.6 and 2.9 years older than their wives.

21.18a.Method 2 would be appropriate because we are not interested in the difference within couples, but rather between the mean for men and the mean for women.

b.The desired interval is 5.1 ± 2(0.26) or 4.58 inches to 5.62 inches. We can be 95% certain that the average height for British men is between 4.58 and 5.62 inches taller than the average height for British women.

21.19a.Neither the women taking the pills or the medical personnel with whom they interacted knew who had which type of pill (calcium or placebo).

b.Because this was a randomized experiment and not an observational study, it is actually possible to make a causal conclusion. Confounding variables, if any, should have been equally present in both groups, so the differences seen between the calcium and placebo-treated groups can be attributed to the calcium.

21.20A study to compare the IQs of children whose mothers smoked and did not smoke during pregnancy found that for the children included in the study, the smokers’ children’s IQs were about 2.59 points lower, after controlling for factors such as mother’s and father’s education. However, the difference for the children in the sample may not reflect a real difference in the population of mothers and babies. Based on the children in the study, all we can say is that we are fairly certain the difference in the population is between −3.03 and +8.20 IQ points. Therefore, it is even possible that the IQs were lower for the children of smokers.

21.21a.The drivers in the sample sometimes were observed reading or writing and thus not to have their hands on the wheel. Taking into account the variability from driver to driver in this sample, we estimate that the average amount of time drivers in the general population spend with their hands not on the wheel because they are reading or writing is probably between 4.24% and 34.39% of all time in the car.

b.The drivers in the sample occasionally were observed to take their hands off the wheel because they were conversing. Taking into account the variability from driver to driver in this sample, we estimate that the average amount of time drivers in the general population spend with their hands not on the wheel because they are conversing is minimal, probably between 0.92% and 2.38% of all time in the car.

c.Multiply the endpoints by 60 minutes to get 2.5 to 20.6 minutes.

d.The two features are the size of the sample and the variability in the measurement of interest.

21.22a.We can be 95% confident that between 0.10% and 0.20% of the population of people similar to the ones in these studies would commit suicide while taking SSRIs.

b.The two intervals have very substantial overlap, so it’s possible that there is no difference in the effectiveness of placebo and SSRIs for preventing suicide.

c.The results are based on randomized experiments, so a cause-and-effect conclusion can be made. It appears that SSRIs are no more effective than placebos in preventing suicides.

21.23a.The interval is 1.06 to 1.37. We can be fairly confident that the risk of dying from all causes during the time this study took place was lower for people attending church at least once a week than for those attending less often or not at all. Based on results from the study, we can be fairly confident that the risk of dying for those not attending church at least once a week ranges from 1.06 to 1.37 times the risk of dying for those who do attend church as least once a week.

b.A relative risk of 1.0 would indicate equal risk.

c.If the confidence interval for relative risk is completely above 1.0, then this indicates that the risk of death in the population is lower for those who attend religious services at least weekly. This holds for circulatory disease.

d.This was an observational study. Thus, it cannot be concluded that attending regular church services causes a change in risk of death.

21.24a.The confidence intervals are Mean ± 2 standard errors. In both cases the standard error is given as 4, so the intervals are Mean ± 8. With sleep restriction: 38.3 ± 8 or 30.3 to 46.3. No sleep restriction: 32.4 ± 8 or 24.4 to 40.4.

b.Twelve men between the ages of 20 and 26, all experienced drivers, slept 5 hours on a given night, then at lunchtime the next day ate two cheese rolls and drank 75 ml of 37.5% proof vodka. About half an hour after drinking, their blood alcohol concentration (BAC) was measured. The average for the 12 men in the study was 38.3 mg alcohol per 100 ml of blood. We are 95% confident that the mean BAC for all men similar to these, under the same conditions, would be between 30.3 and 46.3 mg alcohol/100 ml blood.

c.It would not be appropriate because the same men were used in both conditions. Therefore, the samples are not independent.

21.25a.We are 95% confident that the average cognitive score at age 8 for children who had a low birth weight (0 to 2.50 kg) is between 0.11 and 0.42 lower than the average cognitive score for children whose birth weights were closer to average (3.01 to 3.50 kg). The study was done on babies born in the United Kingdom in 1946, but should extend to babies born under similar conditions elsewhere.

b.A difference of 0 would indicate that the population means are equal.

c.There is a clear difference for any groups where the confidence interval does not cover 0. The low birth weight group clearly has a lower average score than the midrange group and the high-normal birth weight group clearly has a higher average score than the midrange group.

d.A cause and effect conclusion cannot be made because this is obviously an observational study. Birth weight cannot be randomly assigned. There are many potential confounding variables, such as mother’s diet, quality of health care available, and so on.

NOTES ABOUT THE MINI-PROJECTS FOR CHAPTER 21

Mini-Project 21.1

This is a wide-ranging project. Make sure the confidence interval interpretation is correct, and that the interpretation is not made by assuming the interval covers 95% of all population values, which is a common mistake. The project should also identify whether the study is an experiment, an observational study or a survey and consider the appropriate "difficulties and disasters" from Chapters 4 and 5.

Mini-Project 21.2

See the cautions about interpreting a confidence interval given in the notes for Mini-Project 21.1. Also, make sure the project includes a discussion of possible sources of bias, including sampling bias and possible measurement bias.

Mini-Project 21.3

Make sure the two groups or conditions are independent of each other and that they are sampled independently. For example, if the two groups consist of males and females, don't use couples unless you can be sure their measurements would not be correlated. (We have seen that couples do have correlated measurements on things like age and height.) Be sure that the interpretation is done correctly, and that it applies to the difference in the two population means, and not 95% of all possible differences in the measurements themselves. Finally, be sure possible biases are discussed concerning the choice of units to measure and the taking of the measurements themselves.

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