Question

Assume that the marks of students in Statistical Methods are normally distributed. A random sample of 25 students’ marks yields a sample mean of 70 and a sample standard deviation of 10. (a) Estimate the population mean of the marks with 95% confidence.

(b) If the population mean of the marks is 67 and the probability that the sample mean of 25 students’ marks is greater than 70 is 3%, find the population standard deviation of the marks

Answer 1

a) For n - 1 = 24 degrees of freedom, we get from the t distribution tables that:

P( -2.064 < t₂₄ < 2.064 ) = 0.95

Therefore the 95% confidence interval for mean here is obtained as:

This is the required 95% confidence interval for the population mean

b) Here, we are given that:

From central limit theorem, we have the distribution of the sample mean as:

Now, we have:

From standard normal tables, we have:

P(Z > 1.881 ) = 0.03

Therefore, we get the z score here as 1.881

So, we get:

Therefore the population standard deviation of marks is 7.9745