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The given data depicts the annual sunspot numbers (denoted by x) paired with annual high values of the Dow Jones Industrial Average (DJIA, denoted by y) in a tabulated form.

To test the significance of linear correlation between the sunspots and DJIA, the null and alternative hypothesis is formulated as,

\[\begin{aligned}{l}{H_0}:\,\rho = 0\\{H_1}:\,\rho \ne 0\end{aligned}\]

Where \[\rho \] represents the true linear correlation coefficient between populations of variable x and y.

The test statistic with\(n - 2\)degrees of freedom is given by:

\(t = \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\)

Where n is the total number of observations and r is the correlation coefficient of sampled observations.

The value of the correlation coefficient is calculated below:

The correlation coefficient r is given as:

\(r = \frac{{n\sum {xy} - \left( {\sum x } \right)\left( {\sum y } \right)}}{{\sqrt {\left( {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \right)\left( {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} \right)} }}\)

From the above table, substitute all the values in the equation.

\[\begin{aligned}{c}r = \frac{{\left( {10 \times 6269151} \right) - \left( {461 \times 134236} \right)}}{{\sqrt {\left( {\left( {10 \times 22273} \right) - {{461}^2}} \right)\left( {\left( {10 \times 1852612654} \right) - {{134236}^2}} \right)} }}\\ = \frac{{808714}}{{\sqrt {10209 \times 506822844} }}\\ = \frac{{808714}}{{2274676.8}}\\ = 0.356\end{aligned}\]

The correlation coefficient is 0.356.

Now, the test statistic is computed as,

\[\begin{aligned}{c}t = \frac{{0.356}}{{\sqrt {\frac{{1 - {{0.356}^2}}}{{\left( {10 - 2} \right)}}} }}\\ = \frac{{0.356}}{{\sqrt {0.109} }}\\ = \frac{{0.356}}{{0.331}}\\ = 1.076\end{aligned}\]

The degrees of freedom will be as shown below.

\(\begin{aligned}{c}n - 2 = 10 - 2\\ = 8\end{aligned}\)

Use the t-table, the p-value corresponding to 8 degrees of freedomis,

\(\begin{aligned}{c}P - value = 2P\left( {T > 1.076} \right)\\ = 0.313\end{aligned}\)

The decision rule;

• If \(p - value > \alpha \) , then, do not reject the null hypothesis.
• If \(p - value < \alpha \), then, reject the null hypothesis.

Here, p-value is greater than the 0.05 level of significance, then the null hypothesis is failed to be rejected.

It means that there is insufficient evidence to conclude that there is a significant linear correlation between sunspots and the DJIA.

This result shows insignificant correlation between the two variables.

The result is not surprising as the correlation (linear relationship)was not expected between the sunspots numbers and the behavior of stocks.