91Ó°ÊÓ

Socially conscious investors screen out stocks of alcohol and tobacco makers, firms with poor environmental records, and companies with poor labor practices. Some examples of "good," socially conscious companies are Johnson and Johnson, Dell Computers, Bank of America, and Home Depot. The question is, are such stocks overpriced? One measure of value is the \(\mathrm{P} / \mathrm{E}\), or. price-to-earnings, ratio. High \(\mathrm{P} / \mathrm{E}\) ratios may indicate a stock is overpriced. For the S\&P stock index of all major stocks, the mean \(\mathrm{P} / \mathrm{E}\) ratio is \(\mu=19.4\). A random sample of 36 "socially conscious" stocks gave a \(\mathrm{P} / \mathrm{E}\) ratio sample mean of \(\bar{x}=17.9\), with sample standard deviation \(s=5.2\) (Reference: Morningstar, a financial analysis company in Chicago). Does this indicate that the mean \(\mathrm{P} / \mathrm{E}\) ratio of all socially conscious stocks is different (either way) from the mean \(\mathrm{P} / \mathrm{E}\) ratio of the \(S \& P\) stock index? Use \(\alpha=0.05\).

Step by step solution

01

Identify the Hypotheses

We need to set up the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The null hypothesis is that the mean P/E ratio of socially conscious stocks is equal to the mean P/E ratio of the S&P stock index (\(\mu = 19.4\)). The alternative hypothesis is that the mean P/E ratio of socially conscious stocks is different from 19.4 (\(\mu eq 19.4\)). Therefore,\[H_0: \mu = 19.4\]\[H_a: \mu eq 19.4\]

02

Calculate the Test Statistic

We will use the t-test for the sample mean because the population standard deviation is unknown and the sample size is 36. The formula for the test statistic is:\[t = \frac{\bar{x} - \mu}{s / \sqrt{n}}\]Substitute the values:\(\bar{x} = 17.9\), \(\mu = 19.4\), \(s = 5.2\), \(n = 36\).\[t = \frac{17.9 - 19.4}{5.2 / \sqrt{36}} = \frac{-1.5}{5.2 / 6}\]Calculate the value:\[t = \frac{-1.5}{0.8667} \approx -1.73\]

03

Determine the Critical Value and Decision Rule

For a two-tailed test at \(\alpha = 0.05\) and 35 degrees of freedom (\(n-1=35\)), we will use the t-distribution table to find the critical value. The critical t-values are approximately \(±2.030\) at \(df = 35\) for a two-tailed test. The decision rule is: if the calculated t-statistic falls outside the range of -2.030 to 2.030, we reject the null hypothesis.

04

Make Decision

Since the calculated t-statistic \(t = -1.73\) does not fall outside the range of the critical values \(-2.030\) to \(2.030\), we do not reject the null hypothesis. Thus, there is not enough evidence to conclude that the mean P/E ratio of socially conscious stocks is different from 19.4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

t-test

A t-test is a statistical method used to determine if there is a significant difference between the means of two groups. In our case, it helps us check if the average P/E ratio of socially conscious stocks differs from the S&P stock index's average. Since the standard deviation of the entire population is unknown, and we only have a sample size of 36, the t-test is appropriate.

When conducting a t-test, we start by setting up our hypotheses:

• The null hypothesis (\(H_0\)) states that there is no difference between the means (\(\mu = 19.4\)).
• The alternative hypothesis (\(H_a\)) claims that there is a difference (\(\mu eq 19.4\)).

Next, we calculate the t-statistic using the formula:\[t = \frac{\bar{x} - \mu}{s / \sqrt{n}}\]Here, \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

If the resulting t-statistic falls outside the critical values in the t-distribution table, we reject the null hypothesis, indicating a significant difference.

p/e ratio

The P/E ratio, or price-to-earnings ratio, is a vital metric in the investment world, used to evaluate whether a stock is potentially overpriced. It compares a company's current share price to its per-share earnings. A high P/E ratio often suggests that investors expect higher earnings growth in the future compared to companies with a lower P/E ratio.

For socially conscious investors, assessing the P/E ratio helps determine if these ethically chosen stocks perform as well, if not better, in terms of value. In our problem, we're comparing the mean P/E ratio of socially conscious stocks against the standard market average, which is 19.4 according to the S&P index.

If a company's P/E ratio is notably higher than the market average, it might be seen as overpriced. On the other hand, a lower P/E ratio might indicate an undervaluation or lesser growth expectations. Hence, understanding these metrics can guide better investment decisions.

critical value

Critical values are boundaries in hypothesis testing that determine whether to reject the null hypothesis. They are based on the significance level, \(\alpha\), which represents the probability of rejecting the null hypothesis when it is actually true.

In our exercise, we use a two-tailed test with \(\alpha = 0.05\). For the sample size of 36, with degrees of freedom (\(n-1=35\)), we consult the t-distribution table to find critical values. In this case, they are approximately \(\pm 2.030\).

The decision rule in hypothesis testing is straightforward:

• If the calculated t-statistic exceeds these critical values, it indicates that we have enough evidence to reject the null hypothesis.
• If it does not exceed these values, we fail to reject the null, suggesting insufficient evidence of a difference between the groups being compared.

socially conscious investing

Socially conscious investing, also known as ethical investing, is a strategy where investors choose stocks based on societal impact, preferring companies with positive environmental, social, and governance (ESG) practices. Investors typically avoid stocks in industries such as tobacco, alcohol, or those with poor labor practices and environmental records.

Examples of such ethically vetted companies include Johnson and Johnson, Bank of America, and Dell Computers. These companies are often seen as contributing positively to society, rather than just focusing on profit.

For investors, socially conscious investing means balancing ethical considerations with financial returns. As the exercise suggests, examining the P/E ratios of these firms compared to industry standards helps investors understand whether they might be compromising financial returns for ethical considerations. It's a growing field, showing that many investors believe financial success can align with positive societal impact.