Question

Given a sample size of 65, with sample mean 726.2 and sample standard deviation 85.3, we perform the following hypothesis test.

Ho: μ = 750
H1: μ < 750

What is the conclusion of the test at the σ= 0.10 level? Explain your answer.

Answer 1

Conditions

Given a sample size of 65, with sample mean 726.2 and sample standard deviation 85.3, we perform the following hypothesis test.

Ho: $\mu = 750$

H1: $\mu < 750$

What is the conclusion of the test at the $\sigma = 0.10$ level? Explain your answer.

Solution

Here we must use t-test.

$$E(X) = 726.2$$ $$\sigma = 85.3$$

Following to the t-test criterion:

$$t = \frac{|E(X) - \mu_0|}{\sigma / \sqrt{n}} = \frac{|726.2 - 750|}{85.3 / \sqrt{65}} \approx 2.24949$$

The degrees of freedom used in this test is $n - 1 = 64$

So, for these degrees there are following t-criterion (taken from a special table of t-criterion values):

- 1.997 for $p = 0.95$

- 2.3851 for $p = 0.98$

- 2.6536 for $p = 0.01$

- 3.4466 for $p = 0.001$

As we have $t = 2.24949$, which is $> 1.997$ but $< 2.3851$, we can say, that with probability of $0.95\%$ the hypothesis Ho is rejected.

For $\sigma = 0.10$ we have the following calculations:

$$t = \frac{|E(X) - \mu_0|}{\sigma / \sqrt{n}} = \frac{|726.2 - 750|}{0.1 / \sqrt{65}} \approx 1918.82$$

t-criterion = 3.4466 for $p = 0.001$ and no more values for t-criterion in the table.

Here we can say, that with the probability of $99.9\%$ the hypothesis Ho is rejected.