Questions: Researchers wanted to determine if having a bean bag chair in the bedroom is associated with obesity. The researchers asked adolescents. After analyzing the results, the researchers determined that the body mass index of the adolescents who had a bean bag chair in their bedroom was different than that of the adolescents who did not have a bean bag chair in their bedroom. Complete parts (a) through (e) below. What is the explanatory variable? A. The explanatory variable is the body mass index of the adolescents B. The explanatory variable is whether the adolescent has a bean bag chair in the bedroom or not C. The explanatory variable is the number of the adolescents who participate in the study. D. The explanatory variable is the number of the adolescents who have a bean bag chair in their bedroom. Can you think of any lurking variables that may affect the results of the study? A. Yes. For example, a possible lurking variable might be the age of the individual. B. No, there are no lurking variables in this study. C. Yes. For example, possible lurking variables might be eating habits and the amount of exercise per week. D. There is not enough information to answer this question.

Researchers wanted to determine if having a bean bag chair in the bedroom is associated with obesity. The researchers asked adolescents. After analyzing the results, the researchers determined that the body mass index of the adolescents who had a bean bag chair in their bedroom was different than that of the adolescents who did not have a bean bag chair in their bedroom. Complete parts (a) through (e) below. What is the explanatory variable? A. The explanatory variable is the body mass index of the adolescents B. The explanatory variable is whether the adolescent has a bean bag chair in the bedroom or not C. The explanatory variable is the number of the adolescents who participate in the study. D. The explanatory variable is the number of the adolescents who have a bean bag chair in their bedroom. Can you think of any lurking variables that may affect the results of the study? A. Yes. For example, a possible lurking variable might be the age of the individual. B. No, there are no lurking variables in this study. C. Yes. For example, possible lurking variables might be eating habits and the amount of exercise per week. D. There is not enough information to answer this question.
Transcript text: Researchers wanted to determine if having a bean bag chair in the bedroom is associated with obesity. The researchers asked adolescents. After analyzing the results, the researchers determined that the body mass index of the adolescents who had a bean bag chair in their bedroom was different than that of the adolescents who did not have a bean bag chair in their bedroom. Complete parts (a) through (e) below. What is the explanatory variable? A. The explanatory variable is the body mass index of the adolescents B. The explanatory variable is whether the adolescent has a bean bag chair in the bedroom or not C. The explanatory variable is the number of the adolescents who participate in the study. D. The explanatory variable is the number of the adolescents who have a bean bag chair in their bedroom. Can you think of any lurking variables that may affect the results of the study? A. Yes. For example, a possible lurking variable might be the age of the individual. B. No, there are no lurking variables in this study. C. Yes. For example, possible lurking variables might be eating habits and the amount of exercise per week. D. There is not enough information to answer this question.
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Solution

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Solution Steps

Step 1: Standard Error Calculation

The Standard Error \( (SE) \) is calculated using the formula:

\[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{1.225}{10} + \frac{0.5583}{10}} = 0.4223 \]

Step 2: Test Statistic Calculation

The test statistic \( (t) \) is computed as follows:

\[ t = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{24.15 - 21.15}{0.4223} = 7.104 \]

Step 3: Degrees of Freedom Calculation

The degrees of freedom \( (df) \) are calculated using the formula:

\[ df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}} = \frac{0.0318}{0.002} = 15.7929 \]

Step 4: P-value Calculation

The p-value \( (P) \) is determined as follows:

\[ P = 2(1 - T(|t|)) = 2(1 - T(7.104)) = 0.0 \]

Step 5: Summary of Results

The results of the Welch's t-test are summarized as follows:

  • \( t \)-statistic: \( 7.104 \)
  • p-value: \( 0.0 \)
  • degrees of freedom: \( 15.7929 \)
  • critical value: \( 2.1222 \)
Step 6: Explanatory Variable

The explanatory variable in this study is:

\[ \text{The explanatory variable is whether the adolescent has a bean bag chair in the bedroom or not.} \]

Step 7: Lurking Variables

Possible lurking variables that may affect the results include:

\[ \text{Yes. For example, possible lurking variables might be eating habits and the amount of exercise per week.} \]

Final Answer

The explanatory variable is whether the adolescent has a bean bag chair in the bedroom or not, and possible lurking variables include eating habits and exercise.

\(\boxed{\text{Explanatory variable: Bean bag chair presence; Lurking variables: Eating habits, exercise.}}\)

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