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Answer:
Where t follows a t distribution with degrees of freedom and the pooled variance
is given by this formula:
Step-by-step explanation:
Data given
American: 21,17,17,20,25,16,20,16 (Sample 1)
French: 24,18,20,28,18,29,17 (Sample 2)
When we have two independent samples from two normal distributions with equal variances we are assuming that
And the statistic is given by this formula:
Where t follows a t distribution with degrees of freedom and the pooled variance
is given by this formula:
This last one is an unbiased estimator of the common variance
The system of hypothesis on this case are:
Null hypothesis:
Alternative hypothesis:
Or equivalently:
Null hypothesis:
Alternative hypothesis:
Our notation on this case :
represent the sample size for group 1
represent the sample size for group 2
represent the sample mean for the group 1
represent the sample mean for the group 2
represent the sample standard deviation for group 1
represent the sample standard deviation for group 2
First we can begin finding the pooled variance:
And the deviation would be just the square root of the variance:
And now we can calculate the statistic:
Now we can calculate the degrees of freedom given by:
And now we can calculate the p value using the altenative hypothesis:
So with the p value obtained and using the significance level assumed we have
so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 10% of significance we don't have significant differences between the two means.