Deck 21: The Role of Confidence Intervals in Research

Question 1

The formula for a confidence interval for the population mean is based on the rule for sample means, which has assumptions that need to be met.What are those assumptions? (Hint: there are different assumptions depending on whether or not the population is bell-shaped.)

Answer 1

1) if the population is bell-shaped, any size random sample can be taken
2) if the population is not bell-shaped, a large enough sample must be taken (at least 30 measurements).

Question 2

Which of the following statements is not true regarding a 95% confidence interval for the mean of a population?

A)In 95% of all samples, the sample mean will fall within 2 standard errors of the true population mean.
B)95% of the population values will lie within 2 standard errors of the sample mean.
C)In 95% of all samples, the true population mean will be within 2 standard errors of the sample mean.
D)If you add and subtract two standard errors to/from the sample mean, in 95% of all cases you will have captured the true population mean.

Answer 2

95% of the population values will lie within 2 standard errors of the sample mean.

Question 3

Which of the following is the correct way to calculate the standard error of the difference between two (independent) means? Let SEM1 represent the standard error of the first mean, and let SEM2 represent the standard error of the second mean.

A)SEM1 + SEM2
B)SEM1 − SEM2.
C)The square root of (SEM1)2 + (SEM2)2.
D)The square root of (SEM1)2 − (SEM2)2.

Answer 3

The square root of (SEM1)2 + (SEM2)2.

Question 4

For Questions , use the following narrative
Narrative: Male weight loss
Suppose a 95% confidence interval for the average amount of weight loss on a diet program for males is between 13.4 and 18.3 pounds.These results were based on a sample of 42 male participants who were deemed to be overweight at the start of the 4-month study.
{Male weight loss narrative} What is the standard error of the sample mean?

A)4.9
B)2.45
C)1.225
D)None of the above

Question 5

What three items are needed in order to construct a confidence interval to estimate the mean of a population?

Question 6

For Questions , use the following narrative
Narrative: Male weight loss
Suppose a 95% confidence interval for the average amount of weight loss on a diet program for males is between 13.4 and 18.3 pounds.These results were based on a sample of 42 male participants who were deemed to be overweight at the start of the 4-month study.
{Male weight loss narrative} What is the sample mean?

A)2.45
B)13.4
C)15.85
D)18.3

Question 7

For Questions , use the following narrative
Narrative: Classes 1 and 2
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.
{Classes 1 and 2 narrative} What can you conclude?

A)You are confident that the averages for Class 1 and Class 2 are significantly different.
B)You are confident that the average for Class 1 was 7 to 11 points higher than for Class 2.
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.

Question 8

You are taking a class with a large number of students.The professor notes that the midterm exam scores were bell-shaped but doesn't tell you the mean.You take a random sample of 10 students and construct a 95% confidence interval for the mean exam score.Now the professor wants to give everyone 5 extra points on the exam.How will your confidence interval change?

A)The sample mean will increase by 5 points and the margin of error will also.
B)The sample mean will increase by 5 points but the margin of error will stay the same.
C)The sample mean will stay the same but the margin of error will increase by 5 points.
D)Both the sample mean and the margin of error will remain unchanged.

Question 9

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 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.
B)Do a separate confidence interval for each half of the pair and subtract the confidence intervals to get a range of differences.
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.

Question 10

For Questions , use the following narrative
Narrative: Male weight loss
Suppose a 95% confidence interval for the average amount of weight loss on a diet program for males is between 13.4 and 18.3 pounds.These results were based on a sample of 42 male participants who were deemed to be overweight at the start of the 4-month study.
{Male weight loss narrative} What is the margin of error?

A)4.9
B)2.45
C)1.225
D)None of the above

Question 11

For Questions , use the following narrative
Narrative: Cell phone use
How do men and women compare when it comes to talking on the cell phone? Suppose you take a random sample of 100 male cell phone owners and a random sample of 100 female cell phone owners.The average number of minutes for the women per month was 280 with a standard deviation of 10; the average number of minutes for the men per month was 190 with a standard deviation of 30.
{Cell phone use narrative} Suppose the women had only talked for 195 minutes per month, on average, and all the other statistics remained the same.Find the lower and upper limits for the resulting confidence interval (females - males), and interpret the results.Who's talking now?

Question 12

How should you collect your data in order to use it in a confidence interval for the difference of two means?

Question 13

In practice the population standard deviation, which appears in the formula for the standard error of the mean, is not known.Statisticians replace it with the _______________, computed from the data.

Question 14

For Questions use the following narrative
Narrative: Quiz times
The introductory biology class at a large university is taught to hundreds of students each semester.For planning purposes, the instructor wants to find out the average amount of time that students would use to take the first quiz, if they could have as long as necessary to take it.She takes a random sample of 100 students from this population and finds that their average time for taking the quiz is 24 minutes, and the standard deviation is 16 minutes.
{Quiz times narrative} Find a 95% confidence interval for the average time to take this quiz for the whole population of students who take the class.

Question 15

For Questions , use the following narrative
Narrative: Male weight loss
Suppose a 95% confidence interval for the average amount of weight loss on a diet program for males is between 13.4 and 18.3 pounds.These results were based on a sample of 42 male participants who were deemed to be overweight at the start of the 4-month study.
{Male weight loss narrative} Suppose an individual from this population wants to join this diet program, and wants to know how much weight he can expect to lose.What can you say, given the results above?

A)He should expect to lose about 'up to' about 18.3 pounds.
B)He should expect to lose 'up to' about 31.7 pounds.
C)He should expect to lose about 15.85 pounds.
D)He should expect to lose between 13.4 and 18.3 pounds.

Question 16

The standard deviation of all possible sample means (from same sized samples) is called the _______________ of the mean.

Question 17

For Questions , use the following narrative
Narrative: Classes 1 and 2
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.
{Classes 1 and 2 narrative} Suppose you switch the order of Class 1 and Class 2 in your analysis but you 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)Your 95% confidence interval will now be entirely negative: from −7 to −11.
C)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.
D)You are still confident that the classes have significantly different mean scores.

Question 18

Describe a general scenario where you would use a confidence interval for the difference between two means.

Question 19

For Questions use the following narrative
Narrative: Quiz times
The introductory biology class at a large university is taught to hundreds of students each semester.For planning purposes, the instructor wants to find out the average amount of time that students would use to take the first quiz, if they could have as long as necessary to take it.She takes a random sample of 100 students from this population and finds that their average time for taking the quiz is 24 minutes, and the standard deviation is 16 minutes.
{Quiz times narrative} Suppose the professor expects the average time to take the exam is 18 minutes.Do you have enough evidence to say that the professor is wrong in her estimation of the average time to take this quiz? Use statistics to justify your answer.

Question 20

For Questions , use the following narrative
Narrative: Cell phone use
How do men and women compare when it comes to talking on the cell phone? Suppose you take a random sample of 100 male cell phone owners and a random sample of 100 female cell phone owners.The average number of minutes for the women per month was 280 with a standard deviation of 10; the average number of minutes for the men per month was 190 with a standard deviation of 30.
{Cell phone use narrative} Based on these sample results, what is the difference in average time spent on the cell phone for females versus males?

Question 21

For Questions , use the following narrative
Narrative: Side effects
Suppose in a large study investigating side effects of a new drug, it was found that the relative risk of having bad side effects from the new drug (compared to placebo) is 1.3 (with a 95% confidence interval of 1.1 to 1.8).This compares to the relative risk of having bad side effects from the standard drug (compared to placebo), which was found to be 3.4 (with a 95% confidence interval of 1.7 to 7.0), based on large sample results.
{Side effects narrative} These confidence intervals are not in the usual format of 'sample value plus or minus the margin of error' that we have seen.Explain how we can tell this.

Question 22

Suppose you are comparing the proportion of people who successfully stopped smoking on each of two different smoking cessation plans.You randomly assigned subjects to treatment, but you also collected data on each subject on a variety of measurement variables as well.How can this help you to verify that your randomization procedure worked?

Question 23

For Questions , use the following narrative
Narrative: Side effects
Suppose in a large study investigating side effects of a new drug, it was found that the relative risk of having bad side effects from the new drug (compared to placebo) is 1.3 (with a 95% confidence interval of 1.1 to 1.8).This compares to the relative risk of having bad side effects from the standard drug (compared to placebo), which was found to be 3.4 (with a 95% confidence interval of 1.7 to 7.0), based on large sample results.
{Side effects narrative} Does there appear to be a difference in the relative risks of having bad side effects for the new drug, compared to the standard drug? Use confidence intervals to give you a general idea.

Question 24

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)You cannot say, even with 95% confidence that the observed difference in sample means represents a real difference in the population means.
B)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.
C)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.
D)All of the above.

Question 25

Which of the following factors can make it difficult to make correct conclusions about the population based on confidence intervals?

A)Bias in the data collection or sampling process.
B)Confounding variables.
C)Sample sizes that are too small to detect true differences in the population.
D)All of the above.

Question 26

To get the standard error for the difference in the means of two independent samples, you take the square root of the __________ of the squares of the individual standard errors.

Question 27

The difference in the means of two independent samples has __________ variability than the mean of the differences in a matched-pairs sample.(Assume sample sizes are comparable and the values in the pairs are positively correlated.)

Question 28

What does it mean for a confidence interval for the difference of two means to contain zero?

A)You are unable to say there is a difference in the population means.
B)Different samples could give results in either direction; completely above zero, completely below zero, or containing zero.
C)The confidence interval will contain some negative numbers and some positive numbers.
D)All of the above.

Question 29

Journal articles often report means and standard deviations, and they also often report their results in the form (mean +/− standard deviation).Does this latter form represent a real confidence interval for the population mean? If yes, explain why.If not, tell what it does represent.(Assume the populations are bell-shaped.)

Question 30

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 sample(s) selected represent the population(s).
B)That the confidence level is clearly stated.
C)That any real differences for which the researchers imply a causal relationship came from a randomized experiment.
D)All of the above.

Question 31

Are all confidence intervals (whether they be for population means, proportions, or anything else) symmetric around their respective sample values? Explain.

Question 32

For a 95% confidence interval, the value of 95% is called the _______________.

Question 33

Suppose a new drug treatment for headaches is compared to placebo in a two independent group matched-pairs study (measuring difference in pain levels before and after for the placebo group and for the drug group).Suppose the severity of symptoms was substantially reduced for both the placebo and the treatment groups.What should be done next?

A)The placebo group should be ignored from this point on, focusing only on the drug group.
B)The placebo effect should be taken into account by comparing the two groups directly, so we can examine the additional influence of taking the drug on the reduction of symptoms.
C)We should conclude that the drug doesn't do any better job than the placebo does.
D)None of the above.

Question 34

Suppose you want to calculate a confidence interval to estimate the population mean.What happens if you want to be more than 95% confident in your results but you can't take another sample? (Assume data quality is not an issue.)

A)You need a larger confidence level than 95%.
B)You need to use a multiplier that is larger than 2 to create your margin of error.
C)You will have to live with a wider confidence interval.
D)All of the above.

Question 35

You can determine confidence intervals for individual means or for the difference in two means (for large, independent samples) as long as you are given these two things: the means and the __________.

Question 36

For Questions , use the following narrative
Narrative: Side effects
Suppose in a large study investigating side effects of a new drug, it was found that the relative risk of having bad side effects from the new drug (compared to placebo) is 1.3 (with a 95% confidence interval of 1.1 to 1.8).This compares to the relative risk of having bad side effects from the standard drug (compared to placebo), which was found to be 3.4 (with a 95% confidence interval of 1.7 to 7.0), based on large sample results.
{Side effects narrative} How would you have interpreted the results if either of these two confidence intervals had contained the value 1.0?

Question 37

You can determine confidence intervals for individual means or for the difference in two means (for large, independent samples) as long as you are given these three things: the means, the standard deviations, and the __________.

Question 38

If a researchers in an observational study attempts to control for a confounding variable, is it more likely that the resulting confidence interval will be wider or narrower than the confidence interval when he/she didn't control for the confounding variable?

A)It will probably be wider.
B)It will probably be narrower.
C)It will have to be disregarded; you can't adjust or control for confounding variables after the fact.
D)There is no way of knowing.

Question 39

A confidence level is a measure of how much confidence we have that the __________ we used to generate the interval worked.

Question 40

For Questions , use the following narrative
Narrative: Side effects
Suppose in a large study investigating side effects of a new drug, it was found that the relative risk of having bad side effects from the new drug (compared to placebo) is 1.3 (with a 95% confidence interval of 1.1 to 1.8).This compares to the relative risk of having bad side effects from the standard drug (compared to placebo), which was found to be 3.4 (with a 95% confidence interval of 1.7 to 7.0), based on large sample results.
{Side effects narrative} Explain, in words a non-statistics student would understand, what the first confidence interval in this example means.