Q. 70 - The Practice of Statistics for AP

Question

Two-sided test The one-sample t statistic from a sample of $n = 25$ observations for the two-sided test of $H_0 : \mu = 64; H_a : \mu \ne 64$has the value $t =-1.12$.

(a) Find the P-value for this test using (i) Table $B$ and (ii) your calculator. What conclusion would you draw at the $5\%$ significance level? At the $1\%$ significance level?

(b) Redo part (a) using an alternative hypothesis of $H_a : \mu < 64$.

Step-by-Step Solution

Answer

a. Table B: $0.05<p-value<0.1$. Technology: the P-value is $0.2738.$ Fail to reject $H_0$ at both levels.

b. Table B: $0.05<p-value<0.1$. Technology: the P-value is $0.1368$. Fail to reject $H_0$ at both levels.

Step 1: Given information

Two-sided test The one-sample t statistic from a sample of $n = 25$ observations for the two-sided test of $H_0 : \mu = 64; H_a : \mu \ne 64$ has the value$t =-1.12$ .

Step 2: Explanation (part a)

(i) Table $B$: Using Table $B$, we get $0.05<p-value<0.1$

(ii) Calculator: Performing the test with significance level $0.05$,

$p-value: 0.27379726$

Decision: There is not enough evidence to reject $H₀$ at the significance level $0.05$, because your p-value is greater than $0.05$.

Performing the test with significance level $0.1$.

$p-value: 0.27379726$

Decision: There is not enough evidence to reject $H₀$ at the significance level $0.1$, because your p-value is greater than $0.1$.

Conclusion: Fail to reject $H_0$ at both levels.

Step 3: Explanation (part b)

Redo part (a) using an alternative hypothesis of $H_a : \mu < 64$

(i) Table $B$: Using Table $B$, we get $0.05<p-value<0.1$

(ii) Calculator: Performing the test with a significance level$0.05$,

$p-value: 0.13689863$

Decision: There is not enough evidence to reject $H₀$ at the significance level $0.05$, because your p-value is greater than $0.05.$

Performing the test with a significance level $0.1$.

$p-value: 0.13689863$

Decision: There is not enough evidence to reject $H₀$ at the significance level $0.1,$ because your p-value is greater than $0.1$.

Conclusion: Fail to reject $H_0$ at both levels.