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]: \(55-76\) ) describes a survey of randomly selected 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 -0.83 (indicating girls intended, on average, to take fewer science 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

a. The data are paired because each girl provided both the number of science classes she intends to take and the number she thinks boys should take. These two pieces of data are paired for each individual girl. \n b. The 95% confidence interval is calculated to be (-0.83 - 1.96*(1.51/224^0.5), -0.83 + 1.96*(1.51/224^0.5)). This shows that with 95% confidence, the true average difference between the number of science classes girls intend to take and the number they think boys should take is within this interval. The interval is negative, indicating that on average, girls intended to take fewer science classes than they thought boys should take.

Step-by-step solution

Identify Paired Data

In the given data, for each girl, we have two pieces of information: the number of science classes she intends to take and the number she thinks boys her age should take. This means that for each girl, we have a pair of associated data points. Hence, these data are paired.

Calculate the Confidence Interval

The 95% confidence interval can be calculated using the formula: \[ \text{Confidence Interval} = \text{Mean} ± \text{Z-score} * \frac{\text{Standard Deviation}}{\sqrt{\text{Sample size}}} \] Here, the Z-score for a 95% confidence interval is 1.96, the mean of the differences is -0.83, and the standard deviation is 1.51. The sample size is 224 (the number of girls). So the confidence interval is: \[ -0.83 ± 1.96 * \frac{1.51}{\sqrt{224}} \]

Interpret the Confidence Interval

The confidence interval calculated in the previous step gives us a range of values. If the confidence interval contains zero, then the mean difference is not statistically significant. If it does not contain zero, then the difference is statistically significant. In this case, since the interval is negative, it indicates that on average, girls intended to take fewer science classes than they thought boys should take.