Question

The head of institutional research at a university believed that the mean age of​ full-time students was declining. In​ 1995, the mean age of a​ full-time student was known to be 27.4 years. After looking at the enrollment records of all 4934​ full-time students in the current​ semester, he found that the mean age was 27.1​ years, with a standard deviation of 7.3 years. He conducted a hypothesis of Upper H 0​: muequals27.4 years versus Upper H 1​: muless than27.4 years and obtained a​ P-value of 0.0020. He concluded that the mean age of​ full-time students did decline. Is there anything wrong with his​ research?

Answer 1

`t=(27.1-27.4)/((7.3)/(√(4934)))=-2.887`

The degrees of freedom are given by:

`df= n-1 = 4934-1= 4933`

Then the p value for this case calculated as:

`p_v =P(t_(4933)<-2.887) =0.002`

Since the p value is a very lower value using any significance level for example 1% or 5% we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significanctly less than 27.4. So then is not anything wrong with the conclusion

Explanation:

Information provided

`\bar X=27.1` represent the sample mean

`s=7.3` represent the sample standard deviation

`n=4934` sample size

`\mu_o =27.4` represent the value to test

t would represent the statistic

`p_v` represent the p value

System of hypothesis

We want to verify if the mean age of​ full-time students did decline (less than 27.4), the system of hypothesis would be:

Null hypothesis:
`\mu \geq 27.4`

Alternative hypothesis:
`\mu < 27.4`

The statistic for this case is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

Replacing the info we got:

`t=(27.1-27.4)/((7.3)/(√(4934)))=-2.887`

The degrees of freedom are given by:

`df= n-1 = 4934-1= 4933`

Then the p value for this case calculated as:

`p_v =P(t_(4933)<-2.887) =0.002`

Since the p value is a very lower value using any significance level for example 1% or 5% we have enough evidence to reject the null hypothesis and we can conclude that the true mean is significanctly less than 27.4. So then is not anything wrong with the conclusion