Question

Do girls think they don't need to take as many science classes as boys? The article "Intentions of Young Students to Enroll in Science Courses in the Future: An Examination of Gender Differences" (Science Education [1999]: \(55-76\) ) gives information from a survey of children in grades 4,5, and \(6 .\) The 224 girls participating in the survey each indicated the number of science courses they intended to take in the future, and they also indicated the number of science courses they thought boys their age should take in the future. For each girl, the authors calculated the difference between the number of science classes she intends to take and the number she thinks boys should take. a. Explain why these data are paired. b. The mean of the differences was \(-.83\) (indicating girls intended, on average, to take fewer classes than they thought boys should take), and the standard deviation was 1.51. Construct and interpret a \(95 \%\) confidence interval for the mean difference.

Short answer

The data are paired because they come from the same subject: each girl. The 95% confidence interval for the mean difference between the number of science classes each girl plans to take and the number she thinks boys should take can partially be calculated with the given data as -0.83 ± 1.96*1.51. The negative mean difference indicates that girls on average plan to take fewer science classes than they think boys should, but the exact interval cannot be specified without the sample size.

Step-by-step solution

Understanding the Paired Data

Paired data usually involves cases where the data is collected on the same subjects, like here, the number of science classes the girls intend to take and the number they think boys should take. These are related to the same individual girl, thus they form pairs.

Confidence Interval Calculation

The formula to calculate a 95% confidence interval for the mean difference is given by \(\bar{X} ± Z_{\frac{α}{2}} × \frac{s}{\sqrt{n}}\), where \(\bar{X}\) is the sample mean (-0.83 in this case), \(Z_{\frac{α}{2}}\) is the Z score for the desired confidence level (1.96 for a 95% confidence level), \(s\) is the standard deviation (1.51 in this case), and \(n\) is the sample size. Since the sample size is not given, it cannot be incorporated into the formula. However, the other elements can be used to construct the confidence interval as follows: -0.83 ± 1.96*1.51.

Confidence Interval Interpretation

The resulting confidence interval estimates with 95% confidence the population mean difference between the number of science classes the girls intend to take and the number they think boys should take. The negative mean difference indicates that on average, girls intend to take fewer science classes than they think boys should take. However, without the sample size, the precise confidence interval cannot be determined.