Step 1

Given:

\(\displaystyle{\left[{n}={12}\right]}\)

Let us assume:

\(\displaystyle{\left[\alpha={0.05}\right]}\)

Given in the output:

\(\displaystyle{\left[{b}_{{1}}={1.2912}\right]}\)

\(\displaystyle{S}{E}_{{{b}_{{{1}}}}}={0.1347}\)

Determine the hypothesis:

\(\displaystyle{H}_{{0}}:\beta_{{1}}={0}\)

\(\displaystyle{H}_{{0}}:\beta_{{1}}\ne{q}{0}\)

Compute the value of the test statistic:

\(\displaystyle{\left[{t}={\frac{{{b}_{{1}}-\beta_{{1}}}}{{{S}{E}_{{{b}_{{{1}}}}}}}}={\frac{{{1.2912}-{0}}}{{{0.1347}}}}\approx{9.59}\right]}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table B containing the t-value in the row \(\displaystyle{\left[{d}{f}={n}—{2}={12}—{2}={10}:\right]}\)

\(\displaystyle{\left[{P}{<}{2}\times{0.0005}={0.001}\right]}\)</span>

If the P-value is less than or equal to the significance level, then the null

hypothesis is rejected:

\(\displaystyle{\left[{P}{<}{0.05}\Rightarrow{R}{e}{j}{e}{c}{t}{H}_{{0}}\right]}\)</span>

There is sufficient evidence to support the claim that the slope of the population regression line is not zero, which means that the model appears to specify a useful relationship between the two variables.

Result:

Yes.

Given:

\(\displaystyle{\left[{n}={12}\right]}\)

Let us assume:

\(\displaystyle{\left[\alpha={0.05}\right]}\)

Given in the output:

\(\displaystyle{\left[{b}_{{1}}={1.2912}\right]}\)

\(\displaystyle{S}{E}_{{{b}_{{{1}}}}}={0.1347}\)

Determine the hypothesis:

\(\displaystyle{H}_{{0}}:\beta_{{1}}={0}\)

\(\displaystyle{H}_{{0}}:\beta_{{1}}\ne{q}{0}\)

Compute the value of the test statistic:

\(\displaystyle{\left[{t}={\frac{{{b}_{{1}}-\beta_{{1}}}}{{{S}{E}_{{{b}_{{{1}}}}}}}}={\frac{{{1.2912}-{0}}}{{{0.1347}}}}\approx{9.59}\right]}\)

The P-value is the probability of obtaining the value of the test statistic, or a value more extreme. The P-value is the number (or interval) in the column title of Table B containing the t-value in the row \(\displaystyle{\left[{d}{f}={n}—{2}={12}—{2}={10}:\right]}\)

\(\displaystyle{\left[{P}{<}{2}\times{0.0005}={0.001}\right]}\)</span>

If the P-value is less than or equal to the significance level, then the null

hypothesis is rejected:

\(\displaystyle{\left[{P}{<}{0.05}\Rightarrow{R}{e}{j}{e}{c}{t}{H}_{{0}}\right]}\)</span>

There is sufficient evidence to support the claim that the slope of the population regression line is not zero, which means that the model appears to specify a useful relationship between the two variables.

Result:

Yes.