Mean Calculation:
To calculate the mean, sum all the values in the dataset and divide by the number of values (10):
Mean = (85 + 92 + 78 + 89 + 90 + 82 + 75 + 86 + 88 + 91) / 10 = 85.5


P-value Calculation (assuming a one-sample t-test with null hypothesis of population mean = 80):
Step 1: Calculate the standard deviation (SD) of the data: SD = 6.34
Step 2: Calculate the standard error (SE): SE = SD / sqrt(n) = 6.34 / sqrt(10) = 2.01
Step 3: Calculate the t-statistic: t = (sample mean - null hypothesis mean) / SE = (85.5 - 80) / 2.01 = 2.74
Step 4: Look up the p-value associated with the t-statistic and degrees of freedom (n-1 = 9) using a t-distribution table.
Result: p-value ≈ 0.02 (assuming a two-tailed test)


Confidence Interval (CI) Calculation (95% confidence level):
Step 1:
Find the critical t-value for a 95% confidence level with 9 degrees of freedom: t-critical = 2.26
Step 2:
Calculate the margin of error: Margin of Error = t-critical * SE = 2.26 * 2.01 = 4.54
Step 3:
Construct the CI:
Lower bound: Mean - Margin of Error = 85.5 - 4.54 = 80.96
Upper bound: Mean + Margin of Error = 85.5 + 4.54 = 90.04
95% CI:
(80.96, 90.04)


Interpretation:
The mean test score is 85.5.
With a p-value of 0.02, there is evidence to reject the null hypothesis that the population mean is 80 at a 95% confidence level.
We can be 95% confident that the true population mean lies between 80.96 and 90.04.
Note: This example assumes a normal distribution of the data. For large datasets, you can use statistical software to calculate the mean, p-value, and confidence intervals more accurately.