The average age of price is right contestants has increased over the years according to the AARP. The mean age in 1980 was 54 years. A sample of the ages of nine contestants on the show had a mean of 64 a standard deviation of 16 years. At α =0.05, test the claim.

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that mean is higher than 54 at 5% of significance.

Step-by-step explanation:

Data given and notation

[tex]\bar X=64[/tex] represent the sample mean

[tex]s=12[/tex] represent the sample standard deviation

[tex]n=9[/tex] sample size

[tex]\mu_o =54[/tex] represent the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean has increased from 54, the system of hypothesis would be:

Null hypothesis:[tex]\mu \leq 54[/tex]

Alternative hypothesis:[tex]\mu > 54[/tex]

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex] (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".

Calculate the statistic

We can replace in formula (1) the info given like this:

[tex]t=\frac{64-54}{\frac{16}{\sqrt{9}}}=1.875[/tex]

P-value

The first step is calculate the degrees of freedom, on this case:

[tex]df=n-1=9-1=8[/tex]

Since is a one side test the p value would be:

[tex]p_v =P(t_{(8)}>1.875)=0.0488[/tex]

Conclusion

If we compare the p value and the significance level given [tex]\alpha=0.05[/tex] we see that [tex]p_v<\alpha[/tex] so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that mean is higher than 54 at 5% of significance.