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Chapter 11: Problem 42

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

Expert verified
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 卤 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

01

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.
02

Confidence Interval Calculation

The formula to calculate a 95% confidence interval for the mean difference is given by \(\bar{X} 卤 Z_{\frac{伪}{2}} 脳 \frac{s}{\sqrt{n}}\), where \(\bar{X}\) is the sample mean (-0.83 in this case), \(Z_{\frac{伪}{2}}\) 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 卤 1.96*1.51.
03

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.

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Key Concepts

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

Paired Data Analysis
In many research scenarios, we examine data where there are natural pairings or linkages within the data itself. This pattern is termed as **paired data** and is commonly analyzed in statistics. In our exercise, data are considered paired because each comparison is within the same individual. Here, each girl has provided two values: the number of science classes she intends to take herself and the number she believes boys should take.
Therefore, both values are uniquely tied to the same survey respondent, making them paired. By analyzing such data, we can assess the differences within the same subjects, which provides more accurate insights when comparing paired measures than treating them as unrelated.
This approach reduces variability not related to the paired factor (in this case, the individual girls), thus offering a clearer picture of the underlying trend or difference.
Confidence Interval
A **confidence interval** is a range of values we are fairly sure our true value lies within. If we want to understand more about the population mean difference in the girls鈥� intentions versus their thoughts on boys' intentions, constructing a confidence interval is essential.
The interval provides an estimated range of values, which is likely to include the population parameter (in this case, the mean difference), with a stated level of confidence (e.g., 95%). This means we are 95% certain that the population mean difference falls within this interval.
For a 95% confidence interval, we use the formula: - \(\bar{X} 卤 Z_{\frac{伪}{2}} \times \frac{s}{\sqrt{n})\)- \(\bar{X} \) is the sample mean (-0.83 here), - \(Z_{\frac{伪}{2}}\) is the Z-score (1.96 for a 95% CI), - \(s\) is the standard deviation (1.51).Without the sample size (), the exact confidence interval calculation is incomplete. However, using other given data, you can construct a basic interval, indicating the range of potential mean differences.
Mean Difference Calculation
The **mean difference calculation** tells us how much, on average, one paired measure differs from the other. In our study of gender differences in education, we are comparing the number of science courses the girls intend to take against the number they believe boys should take.
To find these differences, subtract the number of intended courses for girls from that for boys: \(X_{boy} - X_{girl}\). This gives the difference for each girl. - Averaging these differences gives us the mean difference, \(\bar{X} = -0.83\), indicating a trend where girls plan to take fewer courses on average.- The negative sign shows the direction of difference: girls, on average, expect themselves to take fewer classes compared to their expectations of boys.This calculation helps frame the girls' perceptions and intentions in education relative to gender expectations, illustrating societal or personal attitudes towards gender roles in education.

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