SOLUTION: Test the claim that the mean age of the prison population in cone city is less than 26 years. Sample data are summarized as n=25, mean= 24.4, s=9.2 and a=.05 This is not for hom

Question 875125: Test the claim that the mean age of the prison population in cone city is less than 26 years. Sample data are summarized as n=25, mean= 24.4, s=9.2 and a=.05
This is not for home just for study...thank you!
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Hypothesis:

H0:
H1:

is the greek lowercase letter mu (population mean).

This is a one-tailed test to the left.

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Decision Criteria:

We reject the null hypothesis H0 if the p value is less than 0.05 (the given significance level alpha)

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Test Statistic:

We are using the T distribution because
a) we do not know the population standard deviation (very common in most cases)
b) n < 30

t = (xbar - mu)/(s/sqrt(n))
t = (24.4 - 26)/(9.2/sqrt(25))
t = -0.8695652173913
t = -0.87

This is a one tailed test to the left, so we want to find the area to the left of t = -0.87.
Due to symmetry, this is the same as finding the area to the right of t = 0.87

Use a table like this one to find the area to the right of 0.87 to be between 0.15 and 0.20
How am I getting this? The sample size is n = 25, so the there are 25 - 1 = 24 degrees of freedom, df = 24.
We look in the df = 24 row and we are looking for 0.87, but unfortunately it's not in the row; however, we know it's between 0.857 and 1.059.
The probabilities for the one tailed test are 0.20 and 0.15 respectively, so that's how I got the area to be between 0.15 and 0.20

Anyways, the area to the right of t = 0.87 is between 0.15 and 0.20
So the area to the left of t = -0.87 is between 0.15 and 0.20

This is the p-value.

The p-value is between 0.15 and 0.20

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Decision:

The p-value is NOT less than 0.05 (the given significance level alpha). Even if the p-value was the smallest it can get (0.15), it is still not even close to being smaller than 0.05.

So we can't reject the null hypothesis H0

We fail to reject the null hypothesis H0

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Interpretation:

We fail to reject the null hypothesis H0.

So we don't have enough significant statistical evidence to prove that the mean age of the prison population in cone city is less than 26 years.

That means we must assume that the mean age of the prison population in cone city is 26 years (until we can statistically prove otherwise).
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