A sample of 64 statistics students at a small college had a mean mathematics ACT score of 28 with a standard deviation of 4. Estimate the mean mathematics ACT score for all statistics students at this college. Give the 95% confidence interval.
95% = mean ± 1.96 SEm
SEm = SD/√n
To estimate the mean mathematics ACT score for all statistics students at the college and determine the 95% confidence interval, we can use the concept of confidence intervals.
The formula for the confidence interval for the mean is:
CI = X̄ ± (Z * σ/√n)
Where:
- CI is the confidence interval
- X̄ is the sample mean
- Z is the Z-score associated with the desired level of confidence (95% in this case)
- σ is the population standard deviation
- n is the sample size
Given the information in the question:
- X̄ = 28 (sample mean)
- σ = 4 (standard deviation)
- n = 64 (sample size)
- Z for a 95% confidence level is approximately 1.96 (from the Z-table)
Now, let's calculate the confidence interval:
CI = 28 ± (1.96 * 4/√64)
CI = 28 ± (1.96 * 4/8)
CI = 28 ± (1.96 * 0.5)
CI = 28 ± 0.98
CI = (28 - 0.98, 28 + 0.98)
CI ≈ (27.02, 28.98)
Therefore, the estimated mean mathematics ACT score for all statistics students at the college is approximately 28, and the 95% confidence interval for the mean mathematics ACT score is (27.02, 28.98).