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Chapter 9: Problem 1

Which of the following involve independent samples? a. Data Set 14 "Oscar Winner Age" in Appendix B includes pairs of ages of actresses and actors at the times that they won Oscars for Best Actress and Best Actor categories. The pair of ages of the winners is listed for each year, and each pair consists of ages matched according to the year that the Oscars were won. b. Data Set 15 "Presidents" in Appendix B includes heights of elected presidents along with the heights of their main opponents. The pair of heights is listed for each election. c. Data Set 26 "Cola Weights and Volumes" in Appendix B includes the volumes of the contents in 36 cans of regular Coke and the volumes of the contents in 36 cans of regular Pepsi.

Short Answer

Expert verified
Option (c) involves independent samples.

Step by step solution

01

Understand the Concept of Independent Samples

Independent samples involve groups that have no connection or pairing between their data points. This means that the data from one group does not influence or relate to the data from the other group.
02

Analyze Option (a)

In Option (a), the ages of actresses and actors are paired according to the year they won their Oscars. Since the data points are matched based on the year, these are not independent samples.
03

Analyze Option (b)

In Option (b), the heights of elected presidents are paired with the heights of their main opponents from the same election. The data points are matched by election year, so these are also not independent samples.
04

Analyze Option (c)

In Option (c), volumes of Coke and Pepsi cans are compared but there is no pairing between the specific cans of Coke and Pepsi. The volumes of Coke cans are collected independently of the volumes of Pepsi cans, which means these are independent samples.
05

Conclusion

After analyzing all options, only the data in Option (c) involves independent samples as there is no connection or matching between the volumes of Coke and Pepsi cans.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept
Independent samples are a key concept in statistics when comparing two separate groups. These groups have no connection between their data points.
In other words, the data from one group does not influence or relate to the data from the other group. Independent samples are essential when you want to ensure that the comparison between two groups is unbiased and valid. For example, if you are comparing the test scores of students from two different classes, their scores should not be influenced by each other, making them independent samples. By understanding independent samples, you can conduct accurate statistical tests and draw reliable conclusions.
headline of the respective core concept
Paired data, on the other hand, involves pairing each data point in one group with a corresponding data point in another group. This pairing creates a direct relationship between the two sets.
For example, in a study measuring the blood pressure of patients before and after taking medication, each patient's blood pressure before treatment is paired with their blood pressure after treatment. This allows for a more detailed analysis within the same subject.
Paired data is useful when the primary interest is in the difference or change within pairs rather than the comparison between independent groups.
headline of the respective core concept
Statistical analysis involves collecting, examining, and interpreting data to uncover patterns, trends, or relationships. It is crucial for validating hypotheses and making data-driven decisions.
There are various methods to analyze data depending on whether you're dealing with independent samples or paired data. For independent samples, t-tests or ANOVA might be used. For paired data, paired t-tests would be appropriate.
Understanding the right statistical method to use based on the type of data ensures accurate and relevant results.
headline of the respective core concept
Data comparison is about evaluating two or more data sets to identify similarities or differences. The comparison can show differences in means, proportions, or variances.
When comparing independent samples, it's important to remember that there's no inherent relationship between the samples鈥� data points. This contrasts with paired data, where each comparison is between matched pairs.
Using appropriate statistical techniques based on whether data is independent or paired allows for meaningful and trustworthy comparisons.

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Most popular questions from this chapter

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal. Listed below are "attribute" ratings made by participants in a speed dating session. Each attribute rating is the sum of the ratings of five attributes (sincerity, intelligence, fun, ambition, shared interests). The listed ratings are from Data Set 18 "Speed Dating" in Appendix B. Use a 0.05 significance level to test the claim that there is a difference between female attribute ratings and male attribute ratings. $$\begin{array}{|l|l|l|l|l|l|l|l|l|l|} \hline \text { Rating of Male by Female } & 29 & 38 & 36 & 37 & 30 & 34 & 35 & 23 & 43 \\ \hline \text { Rating of Female by Male } & 36 & 34 & 34 & 33 & 31 & 17 & 31 & 30 & 42 \\ \hline \end{array}$$

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9 - 1 along with "Table" answers based on Table \(A\) - 3 with df equal to the smaller of \(\boldsymbol{n}_{I}-\boldsymbol{I}\) and \(\boldsymbol{n}_{2}-\boldsymbol{I} .\) ) Color and Creativity Researchers from the University of British Columbia conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given below. Higher scores correspond to more creativity. The researchers make the claim that "blue enhances performance on a creative task." a. Use a 0.01 significance level to test the claim that blue enhances performance on a creative task. b. Construct the confidence interval appropriate for the hypothesis test in part (a). What is it about the confidence interval that causes us to reach the same conclusion from part (a)? $$\begin{array}{l|l} \hline \text { Red Background: } & n=35, \bar{x}=3.39, s=0.97 \\ \hline \text { Blue Background: } & n=36, \bar{x}=3.97, s=0.63 \\ \hline \end{array}$$

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. In a study of treatments for very painful "cluster" headaches, 150 patients were treated with oxygen and 148 other patients were given a placebo consisting of ordinary air. Among the 150 patients in the oxygen treatment group, 116 were free from headaches 15 minutes after treatment. Among the 148 patients given the placebo, 29 were free from headaches 15 minutes after treatment (based on data from "High-Flow Oxygen for Treatment of Cluster Headache," by Cohen, Burns, and Goads by, Journal of the American Medical Association, Vol. \(302,\) No. 22 ). We want to use a 0.01 significance level to test the claim that the oxygen treatment is effective. a. Test the claim using a hypothesis test. b. Test the claim by constructing an appropriate confidence interval. c. Based on the results, is the oxygen treatment effective?

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. (Note: Answers in Appendix D include technology answers based on Formula 9 - 1 along with "Table" answers based on Table \(A\) - 3 with df equal to the smaller of \(\boldsymbol{n}_{I}-\boldsymbol{I}\) and \(\boldsymbol{n}_{2}-\boldsymbol{I} .\) ) Is Old Faithful Not Quite So Faithful? Listed below are time intervals (min) between eruptions of the Old Faithful geyser. The "recent" times are within the past few years, and the "past" times are from 1995. Does it appear that the mean time interval has changed? Is the conclusion affected by whether the significance level is 0.05 or \(0.01 ?\) $$\begin{array}{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l} \hline \text { Recent } & 78 & 91 & 89 & 79 & 57 & 100 & 62 & 87 & 70 & 88 & 82 & 83 & 56 & 81 & 74 & 102 & 61 \\ \hline \text { Past } & 89 & 88 & 97 & 98 & 64 & 85 & 85 & 96 & 87 & 95 & 90 & 95 & & & & & \\ \hline\end{array}$$

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal. A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 15 "Presidents"). a. Use the sample data with a 0.05 significance level to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than \(0 \mathrm{cm} .\) b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)? $$\begin{array}{|l|l|l|l|l|l|l|} \hline \text { Height (cm) of President } & 185 & 178 & 175 & 183 & 193 & 173 \\\ \hline \text { Height (cm) of Main Opponent } & 171 & 180 & 173 & 175 & 188 & 178 \\ \hline \end{array}$$
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