For a  hypothesis comparing two population means, what is the critical value for a one-tailed hypothesis test, using a 5% level of significance level, with both sample sizes equal to 13? Assume the population standard deviations are equal. 

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Assume the gathered data points (`x, y`) from each population respectively are independent and that they are normally distributed.

If the population standard deviation of `x` and `y` is the same and denoted `sigma`, then the sample means `bar x`, `bar y` are such that

`bar x` ~ N(`alpha,sigma/sqrt(13))`   `bar y` ~ N(`beta, sigma/sqrt(13))` where `bar x` and `bar y ` are independent.

To test `alpha > beta` (or `alpha < beta`, but not both of these as it is a one-sided test) against the null `alpha=beta`

the test statistic that we calculate is of the form

`(bar x - bar y)/(SE(bar x - bar y))`

where `SE` is the standard error.

Given that `SE(bar x - bar y) = sqrt(V(bar x) + V(bar y)) = sqrt((2sigma^2)/13) = sqrt(2/13) sigma`

The test statistic is of the form

`sqrt(13/2)((bar x - bar y)/(sigma))`

Now, the 95th percentile of the standard normal distribution is 1.645. The null hypothesis is that `bar x = bar y`. So, if we are testing the alternative `bar x > bar y` at the 5% level, then we check whether our computed test statistic is greater than 1.645. This is the case if

`sqrt(13/2)((bar x - bar y)/sigma) > 1.645`    implies   `bar x - bar y > 1.645 sqrt(2/13)sigma`

implies  `bar x - bar y > 0.645` population standard deviations.

Thus, the two means need to be at least 0.645 population standard deviations apart to conclude at the 5% level of significance that one is larger than the other (pre-specifying beforehand which one we expect to be larger, since the test is one-sided) and reject the null that they are the same.

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial