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.
To estimate the mean mathematics ACT score for all statistics students at the college and calculate the 95% confidence interval, we can use the formula for the confidence interval:
CI = X̄ ± (Z * (σ/√n))
Where:
- CI represents the confidence interval
- X̄ is the sample mean
- Z is the Z-score for the desired confidence level
- σ is the population standard deviation (given as 4)
- n is the sample size (given as 64)
First, we need to find the Z-score for a 95% confidence level. The Z-score can be obtained using statistical tables or with the help of statistical software. For a 95% confidence level, the Z-score is approximately 1.96.
Next, we substitute the given values into the formula:
CI = 28 ± (1.96 * (4 / √64))
Calculating the expression within the parentheses:
CI = 28 ± (1.96 * 0.5)
Now we can calculate the upper and lower limits of the confidence interval:
Upper limit = 28 + (1.96 * 0.5)
Lower limit = 28 - (1.96 * 0.5)
Simplifying the calculations:
Upper limit = 28 + 0.98
Lower limit = 28 - 0.98
Which gives us:
Upper limit = 28.98
Lower limit = 27.02
Therefore, the 95% confidence interval for the mean mathematics ACT score for all statistics students at this college is (27.02, 28.98).
95% = mean ± 1.96 SEm
SEm = SD/√n