Question

You are testing $H_0:\mu =10$ against $H_a:\mu \ne 10$ based on an SRS of $15$

observations from a Normal population. What values of the t statistic are statistically significant at the $\alpha =0.005$ level?

$$\begin{gathered} a.t>3.326 \\ b.t>3.286 \\ c.t>2.977 \\ d.t<-3.326ort>3.326 \\ e.t<-3.286ort>3.286 \end{gathered}$$

Short answer

The correct option is (d) $t<-3.326ort>3.326$

Step-by-step solution

Given information

Sample size $\left(n\right)=15$

Level of significance $\left(\alpha \right)=0.05$

The null and alternative hypotheses are:

$$\begin{gathered} H_0:\mu =10 \\ {H}_a:\mu \ne 10 \end{gathered}$$

Explanation

The degree of freedom is:

$$\begin{gathered} Degreeoffreedom(df)=n-1 \\ =15-1 \\ =14 \end{gathered}$$

The critical value at a $5\%$ significance level and $14$ degrees of freedom can be calculated using the critical value table as follows:

$$\begin{gathered} t_{\alpha /2},df=t_{0.05/2},14 \\ =\pm 3.326 \end{gathered}$$

The test would be significant for the values $t<-3.286$or$t>3.326$

Hence, the correct option is (d).