Question

Suppose that we wish to test whether the population mean of weights of male students at a certain college to be larger than 68 kilograms. Consider α = 0.05 level of significance and assume normal population. We have a random sample with size 10, x = 70 and s^2 = 10.

1. Complete the test by finding out its critical region and draw your conclusion.

2. Complete the test by finding out its p-value and draw your conclusion. Is your conclusion the same with the one in question 1?

3. What’s the probability of having type I error of the test? What’s the probability of having type II error of the test if µ = 69.833?

4. Consider collecting a new set of sample and test again. How many observations do we need in order to get power 0.9 for an alternative with µ = 71.16228?

Answer 1

Given information:

We are testing the claim that population mean of weights of male students at a certain college is larger than 68 kilograms.

Therefore, null hypothesis:

alternative hypothesis:

This is one tailed test.

Test statistic:

Degress of freedom of the test:

df = n - 1 = 10 - 1 = 9

1)

Critical value of 't' for 9 degrees of freedom and level of significance of 0.05 is 1.833.

We reject the null hypothesis if test statistic is greater than 1.833.

Here,

Therefore we reject null hypothesis.

Hence we conclude that there is sufficient evidence to support the claim that population mean of weights of male students at a certain college is larger than 68 kilograms.

2)

We reject null hypothesis if p-value is less than level of significance.

Here,

Therefore we reject null hypothesis.

Hence we conclude that there is sufficient evidence to support the claim that population mean of weights of male students at a certain college is larger than 68 kilograms.

3)

Probability of having type I error is level of significance.

Probability of type II error:

This a right tailed test.

Therefore we fail to reject null hypothesis if t is less than 1.833

We have to find 'x' value such that,

Now we have to find probability of drawing sample mean of less than 69.833 given

Therefore probability of type II error is 0.5

4)

Given information:

power = 0.9

Sample size formula:

where,

And

Therefore,