A school has found from past records that the average scores for boys is 75 in a sample of 45 boys, and the average scores for girls is 85 in a sample of 30 girls. The population standard deviations for boys' and girls' scores were 25 and 17, respectively. The 90% confidence interval for the difference of the population means is approximately (, −2.022). Assume that the distributions of the scores of men and women are normally distributed. Round your answer to two decimal places. Confidence Level Corresponding zc Value
90% confidence zc=1.645
95% confidence zc=1.96
99% confidence zc=2.576

The 90% confidence interval for the difference of the population means is approximately (-17.98, -2.02).

Step-by-step explanation:

Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean and standard deviation

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

Boys:

Mean of 75, sample of 45, standard deviation of 25.

This means that

Girls:

Mean of 85, sample of 30, standard deviation of 17.

This means that

Distribution of the difference of mean grades of boys and girls:

Confidence interval:

As stated, the critical value is

The margin of error is of:

Lower bound:

Upper bound:

The 90% confidence interval for the difference of the population means is approximately (-17.98, -2.02).