Question

The closing prices of two common stocks were recorded for a period of 15 days. The means and variances are $$ \begin{array}{ll} \bar{x}_{1}=40.33 & \bar{x}_{2}=42.54 \\ s_{1}^{2}=1.54 & s_{2}^{2}=2.96 \end{array} $$ a. Do these data present sufficient evidence to indicate a difference between the variabilities of the closing prices of the two stocks for the populations associated with the two samples? Give the \(p\) -value for the test and interpret its value. b. Construct a \(99 \%\) confidence interval for the ratio of the two population variances.

Short answer

Additionally, provide the 99% confidence interval for the ratio of the two population variances. Answer: There isn't enough evidence to suggest significant differences in the variabilities of the closing prices for the two stocks. The 99% confidence interval for the ratio of the two population variances is (0.174, 1.542).

Step-by-step solution

Calculate the F-statistic and Degrees of Freedom

To begin comparing the variances, we need to calculate the F-statistic and the degrees of freedom for each sample. The F-statistic can be calculated using the formula: $$ F = \frac{s_1^2}{s_2^2} $$ where \(s_1^2\) and \(s_2^2\) are the sample variances. Note that we put the larger variance in the numerator to ensure that F is greater than or equal to 1. The degrees of freedom for each sample are one less than the number of observations in the respective samples: $$ df_1 = n_1 - 1 = 15 - 1 = 14 \\ df_2 = n_2 - 1 = 15 - 1 = 14 $$ Using the given sample variances, we have: $$ F = \frac{1.54}{2.96} \approx 0.5203 $$

Calculate the p-value

To find the p-value for our F-test, we can use an F-distribution table or a statistical calculator. Using an F-distribution table, we find the critical value of F at a 0.025 level of significance for both degrees of freedom: $$ F_{0.025} (14, 14) = 2.98 $$ Since our calculated F-statistic (0.5203) is not greater than the critical value (2.98), we fail to reject the null hypothesis, meaning there isn't enough evidence to indicate a difference between the variabilities of the closing prices for the two stocks.

Calculate the Confidence Interval for the Ratio of Variances

To calculate the 99% confidence interval for the ratio of the population variances, we use the following formula: $$ \left(\frac{s_1^2}{s_2^2} \times \frac{1}{F_{\alpha/2}(df_1, df_2)}, \frac{s_1^2}{s_2^2} \times F_{\alpha/2}(df_1, df_2)\right) $$ With our calculated F-statistic (0.5203) and the given critically value (2.98) at a 0.025 level of significance, the confidence interval for the ratio of variances is: $$ \left(\frac{1.54}{2.96} \times \frac{1}{2.98}, \frac{1.54}{2.96} \times 2.98 \right) = \left (0.174, 1.542\right) $$ This means that, with 99% confidence, the true ratio of the population variances lies between 0.174 and 1.542.

Key concepts

F-distribution

The F-distribution is an essential concept for comparing variances between two populations. It arises when you compare two sample variances and want to test if their underlying population variances are equal. This is often done using an F-test, which compares the ratio of two variances.

Key characteristics of the F-distribution include:

• It is right-skewed and always positive, since variance cannot be negative.
• The shape of the distribution is determined by two sets of degrees of freedom, one for the numerator and one for the denominator. In our case, both are 14 as calculated by subtracting one from the number of observations (15-1=14) for each sample.
• In hypothesis testing, the F-statistic calculated from your data is compared to a critical value from the F-distribution table to make decisions.

For the exercise, we calculated an F-statistic of 0.5203. Since this was not greater than our critical value of 2.98, it indicated no significant difference in variances between the two stock prices. This use of the F-distribution is crucial in ensuring that decisions in hypothesis testing are statistically valid.

Population Variance

Variance is a measure of how spread out the values in a data set are around the mean. When we discuss population variance, we are considering the variability of an entire population rather than just a sample. In practice, the true population variance is often unknown, so sample variance is used to estimate it.

Calculating population variance helps us understand the differences and similarities between different groups or datasets. In our exercise, the sample variances for the stocks were 1.54 and 2.96. By comparing these, we assess whether the observed differences are significant regarding the entire population.

Knowing the population variance allows businesses and finance analysts to:

• Evaluate the risk associated with investment in stocks.
• Make informed decisions based on data variability.
• Predict future fluctuations more accurately.

Understanding how variance works, especially when testing hypotheses, improves the robustness of your analysis.

Confidence Interval

A confidence interval provides a range of values that is likely to contain a population parameter, such as variance, with a specific level of confidence. In our exercise, a 99% confidence interval was constructed for the ratio of the two population variances.

Confidence intervals are extremely valuable because:

• They offer a range of plausible values for a population parameter, unlike point estimates which provide a single number.
• They give insights into the precision of an estimate. A narrow interval suggests a more precise estimate.
• Higher confidence levels lead to wider intervals. For example, a 99% confidence interval is broader than a 95% interval, providing greater certainty that the interval contains the parameter.

In the exercise, the confidence interval for the ratio was between 0.174 and 1.542. This means with 99% certainty, the true variance ratio lies within this range. Confidence intervals thus enable us to assess parameter estimates with a known level of risk, critical for sound decision-making.