To find the 90% confidence interval estimate for the average score of all students, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / sqrt(sample size))
First, let's find the critical value for a 90% confidence level.
Since we have a large sample size (81 students), we can use the Z distribution.
Using a confidence level of 90%, we need to find the critical value that leaves 5% of the data in the tails. Since it is a two-tailed test, each tail will have 2.5% of the data.
Consulting the Z-table or using a statistical calculator, we find that the critical value for a 90% confidence level is approximately 1.645.
Now, let's plug in the values we have into the formula:
Confidence Interval = 74.6 ± (1.645) * (11.3 / sqrt(81))
First, we need to calculate the standard error of the mean using the formula: standard deviation / sqrt(sample size):
Standard error = 11.3 / sqrt(81) = 11.3 / 9 = 1.2556 (rounded to 4 decimal places)
Now, we can plug this value into the formula:
Confidence Interval = 74.6 ± (1.645) * (1.2556)
Calculating the upper and lower limits of the confidence interval:
Upper limit = 74.6 + (1.645) * (1.2556) = 74.6 + 2.0659 ≈ 76.66
Lower limit = 74.6 - (1.645) * (1.2556) = 74.6 - 2.0659 ≈ 72.54
Therefore, the 90% confidence interval estimate for the average score of all students is approximately 72.54 to 76.66.