Question

Stocks and Sunspots. Listed below are annual high values of the Dow Jones Industrial Average (DJIA) and annual mean sunspot numbers for eight recent years. Use the data for Exercises 1–5. A sunspot number is a measure of sunspots or groups of sunspots on the surface of the sun. The DJIA is a commonly used index that is a weighted mean calculated from different stock values.

DJIA	14,198	13,338	10,606	11,625	12,929	13,589	16,577	18,054
Sunspot Number	7.5	2.9	3.1	16.5	55.7	57.6	64.7	79.3

Correlation Use a 0.05 significance level to test for a linear correlation between the DJIA values and the sunspot numbers. Is the result as you expected? Should anyone consider investing in stocks based on sunspot numbers?

Short answer

The value of r is 0.731.

Since the test statistic value is greater than the critical value, the null hypothesis is rejected. There is a significant linear correlation between the sunspot numbers and the stock indices.

The results are not what was expected.

Furthermore, no one should examine the sunspot count before investing because the correlation between them is meaningless.

Step-by-step solution

Given information

Data are given on two variables, “DJIA” and “Sunspot Number”.

Hypotheses

The null hypothesis is as follows:

There is no linear correlation between the variables “DJIA” and “Sunspot Number”.

The alternative hypothesis is as follows:

There is a linear correlation between the variables “DJIA” and “Sunspot Number”.

Correlation coefficient

Let x denote the DJIA.

Let y denote the sunspot numbers.

The formula for computing the correlation coefficient (r) between the values of DJIA and sunspot numbers is as follows:

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

The following calculations are done to compute the value of r:

x	y	xy	\({x^2}\)	\({y^2}\)
14198	7.5	106485	201583204	56.25
13338	2.9	38680.2	177902244	8.41
10606	3.1	32878.6	112487236	9.61
11625	16.5	191812.5	135140625	272.25
12929	55.7	720145.3	167159041	3102.49
13589	57.6	782726.4	184660921	3317.76
16577	64.7	1072532	274796929	4186.09
18054	79.3	1431682	325946916	6288.49
\(\sum x \)=110916	\(\sum y \)=287.3	\(\sum {xy} \)=4376942	\(\sum {{x^2}} \)=1579677116	\(\sum {{y^2}} \)=17241.35

Substitute the above values to find r:

\(\begin{aligned} r &= \frac{{n\sum {xy} - \sum x \sum y }}{{\sqrt {n\sum {{x^2}} - {{\left( {\sum x } \right)}^2}} \sqrt {n\sum {{y^2}} - {{\left( {\sum y } \right)}^2}} }}\\ &= \frac{{8\left( {4376942} \right) - \left( {110916} \right)\left( {287.3} \right)}}{{\sqrt {8\left( {1579677116} \right) - {{\left( {110916} \right)}^2}} \sqrt {8\left( {17241.35} \right) - {{\left( {287.3} \right)}^2}} }}\\ &= 0.731\end{aligned}\)

Therefore, the value of r is 0.731.

Test statistic

The test statistic is computed as follows:

\(\begin{aligned} t &= \frac{r}{{\sqrt {\frac{{1 - {r^2}}}{{n - 2}}} }}\;\;\; \sim {t_{\left( {n - 2} \right)}}\\ &= \frac{{0.731}}{{\sqrt {\frac{{1 - {{0.731}^2}}}{{8 - 2}}} }}\\ &= 2.624\end{aligned}\)

Here, n=8.

The degrees of freedom of t are computed as follows:

\(\begin{aligned} df &= n - 2\\ &= 8 - 2\\ &= 6\end{aligned}\)

Using the t-table, the critical value of t for \(\alpha = 0.05\)and df = 6 is 2.4469.

The corresponding p-value is obtained as shown:

\(\begin{aligned} p{\rm{ - value}} &= 2P\left( {T > t} \right)\\ &= 2P\left( {T > 2.624} \right)\\ &= 0.039\end{aligned}\)

Since the test statistic value is greater than the critical value and the p-value is less than 0.05, the null hypothesis is rejected.

Therefore, the linear correlation between the sunspot numbers and DJIA is significant.

Discrepancy from reality

In reality, there is no correlation between stock indices and sunspot numbers.

But, according to the obtained value of r, the correlation comes out to be significant.

Thus, the results differ from what was expected.

Moreover, nobody should consider the sunspot numbers before investing as, in the real sense, the correlation between them does not hold any meaning.