Question

The null and alternate hypotheses are: H0: ????1 ???? ????2 H1: ????1 ???? ????2 A random sample of 15 observations from the first population revealed a sample mean of 350 and a sample standard deviation of 12. A random sample of 17 observations from the second population revealed a sample mean of 342 and a sample standard devi- ation of 15. At the .10 significance level, is there a difference in the population means?

Answer 1

`t=\frac{(350-342)-0}{\sqrt{(12^2)/(15)+(15^2)/(17)}}}=1.674`

The degreess of freedom are given by:

`df = 15+12-2 =25`

Now we can calculate the p value with the following probability:

`p_v =2*P(t_(25)>1.674)=0.107`

For this case since the p value is higher than the significance level we can FAIL to reject the null hypothesis and we can conclude that the true means are NOT significantly different at 10% of significance.

Explanation:

Information provided

`\bar X_(1)=350` represent the mean for sample 1

`\bar X_(2)=342` represent the mean for sample 2

`s_(1)=12` represent the sample standard deviation for 1

`s_(2)=15` represent the sample standard deviation for 2

`n_(1)=15` sample size for the group 2

`n_(2)=17` sample size for the group 2

`\alpha=0.1` Significance level provided

t would represent the statistic

Hypothesis to test

We want to verify if the true means for this case are significantly different, the system of hypothesis would be:

Null hypothesis:
`\mu_(1)-\mu_(2)=0`

Alternative hypothesis:
`\mu_(1) - \mu_(2)\\eq 0`

The statistic is given by:

`t=\frac{(\bar X_(1)-\bar X_(2))-\Delta}{\sqrt{(\sigma^2_(1))/(n_(1))+(\sigma^2_(2))/(n_(2))}}` (1)

And the degrees of freedom are given by
`df=n_1 +n_2 -2=15+17-2=30`

Replacing the info given we got:

`t=\frac{(350-342)-0}{\sqrt{(12^2)/(15)+(15^2)/(17)}}}=1.674`

The degreess of freedom are given by:

`df = 15+12-2 =25`

Now we can calculate the p value with the following probability:

`p_v =2*P(t_(25)>1.674)=0.107`

For this case since the p value is higher than the significance level we can FAIL to reject the null hypothesis and we can conclude that the true means are NOT significantly different at 10% of significance.