Step 1

Hypothesis:

\(\displaystyle{H}_{{{0}}}:\sigma^{{{2}}}={0.18}\)

\(\displaystyle{H}_{{{1}}}:\sigma^{{{2}}}\ne{0.18}\)

Test Statistic \(\displaystyle{\left({X}^{{{2}}}\right)}={\frac{{{\left({n}-{1}\right)}{s}^{{{2}}}}}{{\sigma^{{{2}}}}}}\)

\(\displaystyle={\frac{{{\left({101}-{1}\right)}{\left({0.13}\right)}}}{{{\left({0.18}\right)}}}}\)

\(\displaystyle={72.2222}\)

Thus, the test statistic is 72.2222.

Step 2

P-value is calculated using chi-square test statistic by p-value calculator.

\(\displaystyle{P}-{v}{a}{l}{u}{e}={0.0328096}\)

\(\displaystyle={0.0328}\)

Since P-value is less than 0.05 level of significance, thus there is sufficient evidence to reject the null hypothesis. So \(\displaystyle\sigma^{{{2}}}\ne{0.18}\).

Hypothesis:

\(\displaystyle{H}_{{{0}}}:\sigma^{{{2}}}={0.18}\)

\(\displaystyle{H}_{{{1}}}:\sigma^{{{2}}}\ne{0.18}\)

Test Statistic \(\displaystyle{\left({X}^{{{2}}}\right)}={\frac{{{\left({n}-{1}\right)}{s}^{{{2}}}}}{{\sigma^{{{2}}}}}}\)

\(\displaystyle={\frac{{{\left({101}-{1}\right)}{\left({0.13}\right)}}}{{{\left({0.18}\right)}}}}\)

\(\displaystyle={72.2222}\)

Thus, the test statistic is 72.2222.

Step 2

P-value is calculated using chi-square test statistic by p-value calculator.

\(\displaystyle{P}-{v}{a}{l}{u}{e}={0.0328096}\)

\(\displaystyle={0.0328}\)

Since P-value is less than 0.05 level of significance, thus there is sufficient evidence to reject the null hypothesis. So \(\displaystyle\sigma^{{{2}}}\ne{0.18}\).