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

z Scores Using only the sunspot numbers, identify the highest number and convert it to a z score. In the context of these sample data, is that highest value “significantly high”? Why or why not?

Short answer

The highest sunspot number is equal to 79.3.

The corresponding z-score is equal to 1.38.

As the value of 79.3 lies within the 2 standard deviations of the mean value, the highest value equal to 79.3 is not significantly high.

Step-by-step solution

Given information

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

Maximum value

The highest value of the sunspot numbers given in the table is equal to 79.3.

Conversion to a z-score

The following formula is used to convert a sample observation to a z-score:

\(z = \frac{{x - Mean}}{{S.D.}}\)

The mean value is computed below:

\(\begin{aligned} Mean &= \frac{{7.5 + 2.9 + ..... + 79.3}}{8}\\ &= 35.91\end{aligned}\)

The value of the standard deviation is computed below:

\(\begin{aligned} S.D. &= \sqrt {\frac{{\sum\limits_{i = 1}^n {{{({x_i} - \bar x)}^2}} }}{{n - 1}}} \\ &= \sqrt {\frac{{{{\left( {7.5 - 35.91} \right)}^2} + {{\left( {2.9 - 35.91} \right)}^2} + ...... + {{\left( {79.3 - 35.91} \right)}^2}}}{{8 - 1}}} \\ &= 31.45\end{aligned}\)

Thus, the value of the z-score is computed as follows:

\(\begin{aligned} z &= \frac{{x - Mean}}{{S.D.}}\\ &= \frac{{79.3 - 35.91}}{{31.45}}\\ &= 1.38\end{aligned}\)

Significance of the value

According to the rule, if the sample value lies within 2 standard deviations of the mean value, it is considered insignificant; otherwise, not.

Mathematically, the value of should lie within the following interval to be considered insignificant:

\(\begin{aligned} \left( {\mu - 2\sigma ,\mu + 2\sigma } \right) &= \left( {35.91 - 2\left( {31.45} \right),35.91 + 2\left( {31.45} \right)} \right)\\ &= \left( { - 26.99,98.81} \right)\end{aligned}\)

Since the value of the sunspot number (79.3) lies in the interval, thus the value is not significant.

Therefore, the value of the sunspot number equal to 79.3 is not significantly high.