One of your peers claims that boys do better in math classes than girls. together you run two independent simple random samples and calculate the given summary statistics of the boys and the girls for comparable math classes

High school
1

one of your peers claims that boys do better in math classes than girls. together you run two independent simple random samples and calculate the given summary statistics of the boys and the girls for comparable math classes. in calculus, 15 boys had a mean percentage of 82.3 with standard deviation of 5.6 while 12 girls had a mean percentage of 81.2 with standard deviation of 6.7. what assumptions need to be made in order to determine the 90% confidence interval for the difference in the mean percentage scores for the boys in calculus and the girls in calculus? supposing the assumption is true, calculate the interval. @aibot

2

To determine the 90% confidence interval for the difference in the mean percentage scores, the following assumptions need to be made:

  1. Independence: The two samples of boys and girls must be independent of each other. This assumes that the selection of one student does not influence the selection of another.

  2. Random Sampling: The samples of boys and girls should be selected randomly from the population of all students taking calculus. This ensures that the samples are representative of the entire population.

  3. Normality: The distribution of the scores in each population (boys and girls) should be approximately normal. This assumption is required for the calculation of confidence intervals using t-distribution.

Given the assumption that these assumptions are met, we can calculate the confidence interval:

The point estimate of the difference in mean percentage scores for boys and girls in calculus is:
Point estimate = Mean percentage score for boys - Mean percentage score for girls
= 82.3 - 81.2
= 1.1

Now, to calculate the confidence interval, we need to consider the standard deviations and the sample sizes for both groups.

The standard error of the difference in mean percentage scores is calculated as:
Standard Error = sqrt[(Standard deviation of boys)^2/Number of boys + (Standard deviation of girls)^2/Number of girls]
= sqrt[(5.6^2/15) + (6.7^2/12)]
= sqrt[1.2987 + 3.24694]
= sqrt[4.54564]
= 2.13

Using the t-distribution with (15+12-2) = 25 degrees of freedom and a 90% confidence level (alpha = 0.10), we find the t-value (using a t-table or calculator) to be approximately 1.711.

To calculate the 90% confidence interval for the difference in mean percentage scores for boys and girls in calculus, we use the formula:
Confidence interval = Point estimate +/- (t-value * Standard error)
= 1.1 +/- (1.711 * 2.13)
= 1.1 +/- 3.6466
= (-2.5466, 4.7466)

Therefore, the 90% confidence interval for the difference in the mean percentage scores for boys and girls in calculus is (-2.5466, 4.7466).