Question

A university is interested in promoting graduates of its honors program by establishing that the mean GPA of these graduates exceeds 3.50. A sample of 36 honors students is taken and is found to have a mean GPA equal to 3.60. The population standard deviation is assumed to equal 0.40. To establish whether the mean GPA exceeds 3.50, the appropriate hypotheses are____________

Answer 1

H0:
`\mu \leq 3.5`

H1:
`\mu > 3.5`

Explanation:

1) Data given and concepts involved

n=36 the sample size

`\bar X=3.60` sample mean obtained

`\sigma =0.40`

The null hypothesis "Is what we attempt to find evidence against in our hypothesis test". And is denoted by H0

The alternative hypothesis "Is what we are attempting to demonstrate". and is denoted by H1.

In our case since we want to see if
`\mu > 3.5` then that would be the alternative hypothesis, and the null hypothesis it's always the complement for the alternative hypothesis, so on this case
`\mu \leq 3.5`

The distribution for
`\bar X` is given by:

`\bar X \sim N(\mu,(\sigma)/(√(n)))`

2) Test the hypothesis

In order to test the hypothesis we can use the following statistic:

`z=(\bar X -\mu)/((\sigma)/(√(n)))`

Replacing the values given we have:

`z=(3.6 -3.5)/((0.4)/(√(36)))=1.5`

We know that
`z \sim N(0,1)` and we can calculate a critical value, but we don't know the significance level for the hypothesis, so on this case we can't proceed.