Question

A teacher was interested in the mathematical ability of graduating high school seniors in her state. She gave a 32-item test to a random sample of 75 seniors from her state with the following results:The ΣX=1275 and ΣX²=23525. Establish a 95% confidence interval about the mean, and write a sentence that explains the interval you found to a high school principal.

Answer 1

The 95% confidence interval for the mean mathematical ability score of graduating high school seniors in the state is (16.20, 18.20).

Step-by-step explanation:

The formula for calculating the confidence interval for the mean (\
`(\mu\))`is given by:

`\[ \bar{X} \pm \left( (t * s)/(√(n)) \right) \]`

`- \(\bar{X}\)` is the sample mean,

- t is the t-score corresponding to the desired confidence level and degrees of freedom,

- s is the sample standard deviation,

- n is the sample size.

In this case, the sample mean
`(\(\bar{X}\))` is calculated as
`\((\Sigma X)/(n) = (1275)/(75) = 17\).`

The degrees of freedom df for a sample of 75 is 74. Using a t-table, the critical t-value for a 95% confidence interval is approximately 2.013.

The standard deviation s is calculated using the formula
`(\sqrt{\frac{\Sigma X^2 - n \bar{X}^2}{n-1}}\),` resulting in
`\(s \approx 0.98\).`

Substituting these values into the formula, the confidence interval is calculated as (16.20, 18.20).

This means that we are 95% confident that the true mean mathematical ability score of graduating high school seniors in the state falls within the interval (16.20, 18.20). It provides a range within which we can reasonably estimate the population mean. This information is crucial for the high school principal in understanding the precision of the assessment and making informed decisions based on the students' mathematical abilities.