Q. 69 - The Practice of Statistics for AP

Question

One-sided test Suppose you carry out a significance test of $H_0:\mu =5$ versus $H_a:\mu >5$ based on a sample of size $n = 20$ and obtain $t = 1.81$.

(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 $H_a:\mu \ne 5$.

Step-by-Step Solution

Answer

a.  Table $B$: $0.025 < P-value < 0.05$. Technology: the P-value is $0.043$. Reject $H_0$ at the $5\%$ significance level. Fail to reject $H_0$ at the $1\%$ significance level.

b.  Table $B$: $0.05 < P-value < 0.10$. Technology: the P-value is $0.086$. Fail to reject $H_0$at both levels.

Step 1: Given information

(One-sided test Suppose you carry out a significance test of  $H_0:\mu =5$ versus $H_a:\mu >5$ based on a sample of size $n = 20$and obtain $t = 1.8$.

Step 2: Explanation (part a)

(i) Table $B$:

The degrees of freedom required for looking Table $B$: is given by $df=n-1=20-1=19$

Now for a one tailed significance t test looking at Table $B$: the p-value is calculated as

$0.025 < P-value < 0.05$

(ii) Calculator: 

Performing test on significance level $0.05$: $p-value: 0.04388018$

Decision: You can reject$H₀$ at the significance level $0.05$, because your p-value does not exceed $0.05$.

Performing test on significance level $0.1:$ $p-value: 0.9561198$

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: Reject $H_0$ at the $5\%$ significance level. Fail to reject $H_0$ at the $1\%$ significance level.

Step 3: Explanation (part b)

Redo part (a) using an alternative hypothesis of $H_a: \mu \ne 5$

(i) Table $B$:

The degrees of freedom required for looking Table $B$: is given by

$df=n-1=20-1=19$

Now for a one tailed significance t test looking at Table $B$: the p-value is calculated as

$0.05 < P-value < 0.1$

(ii) Calculator: 

Performing test on significance level $0.05$: $p-value : 0.08776036$

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 test on significance level $0.1$: $p-value: 0.08776036$

Decision: You can reject $H₀$ at the significance level $0.1$, because your p-value does not exceed $0.1$.

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