Problem 5 - Basic Statistics for Business and Economics with Student

Question

A financial analyst wants to compare the turnover rates, in percent, for shares of oil-related stocks versus other stocks, such as GE and IBM. She selected 32 oil-related stocks and 49 other stocks. The mean turnover rate of oil-related stocks is 31.4 percent and the population standard deviation 5.1 percent. For the other stocks, the mean rate was computed to be 34.9 percent and the population standard deviation 6.7 percent. Is there a significant difference in the turnover rates of the two types of stock? Use the .01 significance level.

Step-by-Step Solution

Answer

Yes, there is a significant difference at the 0.01 level.

Step 1: State the Hypotheses

Define the null and alternative hypotheses for the test. The null hypothesis (H0) states that there is no difference in the turnover rates of the two types of stocks. Mathematically, it is expressed as \( H_0: \mu_1 = \mu_2 \), where \( \mu_1 \) is the mean turnover rate of oil-related stocks and \( \mu_2 \) is the mean turnover rate of other stocks. The alternative hypothesis (H1) states there is a difference, expressed as \( H_1: \mu_1 eq \mu_2 \).

Step 2: Determine the Significance Level

The significance level for this test is given as \( \alpha = 0.01 \). This means we have a 1% chance of incorrectly rejecting the null hypothesis if it is actually true.

Step 3: Calculate the Test Statistic

Since population standard deviations are known, use the Z-test for comparison. The formula for the Z-test is:\[Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\]where \( \bar{x}_1 = 31.4 \), \( \bar{x}_2 = 34.9 \), \( \sigma_1 = 5.1 \), \( \sigma_2 = 6.7 \), \( n_1 = 32 \), and \( n_2 = 49 \). Substituting these values gives:\[Z = \frac{31.4 - 34.9}{\sqrt{\frac{5.1^2}{32} + \frac{6.7^2}{49}}} \approx \frac{-3.5}{\sqrt{0.8125 + 0.9166}} \approx \frac{-3.5}{1.301} \approx -2.69\]

Step 4: Find the Critical Value

For a two-tailed test with \( \alpha = 0.01 \), look up the critical values in a Z-table. The critical Z-values are approximately \( \pm 2.576 \) for a .01 significance level split into two tails.

Step 5: Compare the Test Statistic with Critical Value

Compare the calculated Z value of -2.69 to the critical values of \( \pm 2.576 \). Since \(-2.69 < -2.576\), the test statistic falls in the rejection region of the null hypothesis.

Step 6: Make a Decision

Since the calculated Z value is less than the critical Z value, we reject the null hypothesis. This indicates that there is a statistically significant difference between the turnover rates of oil-related stocks and other stocks at the 0.01 significance level.

Key Concepts

Z-test

The Z-test is a statistical method used to determine whether there is a significant difference between the means of two populations. It is particularly useful when the population standard deviations are known and the sample size is large. In our exercise, the financial analyst wants to test if there's a significant difference between the turnover rates of oil-related and other stocks. The Z-test helps in making this decision effectively.

To apply the Z-test, one needs to know the sample means, population standard deviations, and the sample sizes. The formula used is:\[Z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\]where:- \(\bar{x}_1\) and \(\bar{x}_2\) are the sample means.
- \(\sigma_1\) and \(\sigma_2\) are the population standard deviations.
- \(n_1\) and \(n_2\) are the sample sizes.

This calculation helps determine the Z-score, which can then be compared to a critical value to establish statistical significance.

Significance Level

The significance level, denoted by \(\alpha\), is a threshold set by the researcher to decide whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is wrongly rejected. In our exercise, the analyst has chosen a significance level of 0.01 or 1%.

This low significance level indicates that the analyst is very cautious and wants to minimize the risk of incorrectly claiming a difference when none exists. At a 0.01 level, the critical Z-values for a two-tailed test are approximately \(\pm 2.576\). This means if the Z-score falls beyond these values, the null hypothesis is rejected.

Choosing a significance level depends largely on the context and the danger of making incorrect inferences. At 0.01, there's only a 1% chance of false positives, making it a rigorous test criterion for the analysis.

Population Standard Deviation

Population standard deviation is a measure of the spread or dispersion of a set of data. It gives insight into how much the individual data points differ from the population mean. In hypothesis testing, knowing the population standard deviations allows us to use the Z-test effectively.

In the example, the population standard deviations for oil-related stocks and other stocks are 5.1% and 6.7% respectively. These values are crucial in helping calculate the Z-statistic. They show the variability within each population and contribute to the standard error calculation in the Z-test formula.

Understanding these deviations is important because they provide a basis for the spread within the populations being compared. A larger standard deviation implies greater variability, which affects the confidence and reliability of the sample mean as a representation of the population mean.