Question

Explain exactly how a paired $t$-test can be formulated as a onemean$t$-test. (Hint: Work solely with the paired-difference variable.)

Short answer

The null hypothesis is rejected since the numbers fall within the critical range, indicating that people prefer Nibbles to Wribbles.

Step-by-step solution

Given Information

The null and alternate hypotheses are as follows:

$H_0:{\mu }_1={\mu }_2$

$H_a:{\mu }_1\ne {\mu }_2$

Explanation

The standard deviation is,

$s_d=\sqrt{\frac{(x-\overline{x}{)}^2}{n-1}}$

$=\sqrt{\frac{(5-(-1.40){)}^2+(-4-(-1.40){)}^2+\dots +(-3-(-1.40){)}^2}{20-1}}$

$=3.19$

The test statistics is,

$t=\frac{\overline{d}}{\frac{s_d}{\sqrt{n}}}$

Substitute the given values in above equation.

$t=\frac{-1.40}{\frac{3.19}{\sqrt{20}}}$

$=-1.97$

Explanation

The degree of freedom is,

$\text{dof}=n-1$$=20-1$$=19$

The critical value for the level of significance of $0.05$ is $1.729$.