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Ch. 9 - Inferences from Two Samples
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 9 - Inferences from Two SamplesProblem 9.CR.4
Chapter 9, Problem 9.CR.4

In Exercises 1–10, based on the nature of the given data, do the following:


a. Pose a key question that is relevant to the given data.
b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).
c. Analyze the data and state a conclusion.


IQ Scores of Twins Listed below are IQ scores of twins listed in Data Set 12 “IQ and Brain Size” in Appendix B. The data are pairs of IQ scores from ten different families.


"IQ scores of twins from 10 families, comparing first born and second born children."

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1
Step 1: Pose a key question relevant to the data. For example, 'Is there a significant correlation between the IQ scores of first-born twins and second-born twins?' This question helps us understand the relationship between the two sets of scores.
Step 2: Identify a statistical procedure or tool to address the key question. In this case, we can use the Pearson correlation coefficient to measure the strength and direction of the linear relationship between the two sets of IQ scores.
Step 3: Organize the data into pairs for analysis. Each pair consists of the IQ score of the first-born twin and the corresponding IQ score of the second-born twin. For example, the first pair is (96, 89), the second pair is (87, 87), and so on.
Step 4: Apply the formula for the Pearson correlation coefficient: \( r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \cdot \sum (y_i - \bar{y})^2}} \), where \( x_i \) and \( y_i \) are the individual scores, and \( \bar{x} \) and \( \bar{y} \) are the means of the first-born and second-born scores, respectively.
Step 5: Analyze the result of the correlation coefficient. If \( r \) is close to 1 or -1, it indicates a strong positive or negative correlation, respectively. If \( r \) is close to 0, it indicates little to no linear relationship. Based on this, state a conclusion about the relationship between the IQ scores of first-born and second-born twins.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Descriptive Statistics

Descriptive statistics summarize and describe the main features of a dataset. This includes measures such as mean, median, mode, and standard deviation, which provide insights into the central tendency and variability of the data. In the context of the IQ scores of twins, descriptive statistics can help compare the performance of first-born and second-born children.
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Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample of data to support a particular hypothesis about a population. In this case, one might hypothesize that there is a significant difference in IQ scores between first-born and second-born twins. This involves setting up null and alternative hypotheses and using statistical tests to analyze the data.
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Step 1: Write Hypotheses

Correlation and Regression

Correlation and regression analysis are used to examine the relationship between two variables. In this scenario, one could analyze the correlation between the IQ scores of first-born and second-born twins to see if there is a pattern or trend. Regression analysis could further help in predicting IQ scores based on the birth order, providing deeper insights into the data.
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Related Practice
Textbook Question

Test Values p_cap1, p_cap2. Find the values of and the pooled proportion p_bar obtained when testing the claim given in Exercise 1.

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Textbook Question

In Exercises 1–10, based on the nature of the given data, do the following:


a. Pose a key question that is relevant to the given data.

b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).

c. Analyze the data and state a conclusion.



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Textbook Question

Identifying Hypotheses In a randomized clinical trial of adults with an acute sore throat, 288 were treated with the drug dexamethasone and 102 of them experienced complete resolution; 277 were treated with a placebo and 75 of them experienced complete resolution (based on data from “Effect of Oral Dexamethasone Without Immediate Antibiotics vs Placebo on Acute Sore Throat in Adults,” by Hayward et al., Journal of the American Medical Association). Identify the null and alternative hypotheses corresponding to the claim that patients treated with dexamethasone and patients given a placebo have the same rate of complete resolution.

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Textbook Question

Variation Find the value of the test statistic used for testing the claim that the two samples from Exercise 5 are from populations having the same variation.

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Textbook Question

In Exercises 1–10, based on the nature of the given data, do the following:


a. Pose a key question that is relevant to the given data.

b. Identify a procedure or tool from this chapter or the preceding chapters to address the key question from part (a).

c. Analyze the data and state a conclusion.



Video Games In a survey of subjects aged 18–29, subjects were asked if they play video games often or sometimes. Among 984 females, 49% answered “yes” (based on data from a Pew Research Center survey).

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Textbook Question

Sampling Methods A student obtains a sample of responses to the question “Do you plan to take or have you taken a statistics course?” A second student obtains a sample of responses to the same question. The first student surveys only males at the same college, and the second student surveys only females at the same college. What is wrong with the samples? Can randomization be used to overcome the flaws of those samples?

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