Question

a paired t - test was performed on the following hypothesis.

$h_0:\mu_1 = \mu_2$

$h_1:\mu_1\
eq\mu_2$

a test statistic of 2.332 was calculated with a degrees of freedom of 20. use the appropriate table to find the p - value for this hypothesis test.

$\square

Explanation:

Step1: Identify Test Type and Tail

This is a two - tailed paired t - test (since \(H_1:\mu_1
eq\mu_2\)). We have a test statistic \(t = 2.332\) and degrees of freedom \(df=20\).

Step2: Use t - distribution Table

For a two - tailed test, we look at the t - distribution table for \(df = 20\). We find the critical values corresponding to different significance levels.

• For \(df = 20\), the critical value for \(\alpha/2=0.025\) (two - tailed) is \(t_{0.025,20}=2.086\) and for \(\alpha/2 = 0.01\) (two - tailed) is \(t_{0.01,20}=2.528\).
• Since our test statistic \(t = 2.332\) is between \(t_{0.025,20}=2.086\) and \(t_{0.01,20}=2.528\), the p - value (for two - tailed test, the p - value is the probability of getting a \(|t|\) more extreme than the calculated \(t\)) will be between \(2\times0.01 = 0.02\) and \(2\times0.025=0.05\) (because for a two - tailed test, \(p - value=2\times P(T > |t|)\) when \(H_0\) is true, and we use the one - tailed probabilities from the t - table and double them).

Answer:

\(0.02 < p\text{-value}<0.05\)