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Stock fund managers are investment professionals who decide which stocks should be part of a portfolio. In a recent article in the Wall Street Journal ( \(N\) ot \(a\) Stock-Picker's Market, WSJ January 25,2014 ), the performance of stock fund managers was considered based on dispersion in the market. In the stock market, risk is measured by the standard deviation rate of return of stock (dispersion). When dispersion is low, then the rate of return of the stocks that make up the market are not as spread out. That is, the return on Company \(X\) is close to that of \(Y\) is close to that of \(Z,\) and so on. When dispersion is high, then the rate of return of stocks is more spread out; meaning some stocks outperform others by a substantial amount. Since \(1991,\) the dispersion of stocks has been about \(7.1 \% .\) In some years, the dispersion is higher (such as 2001 when dispersion was \(10 \%)\), and in some years it is lower (such as 2013 when dispersion was \(5 \%)\). So, in 2001 , stock fund managers would argue, one needed to have more investment advice in order to identify the stock market winners, whereas in 2013 , since dispersion was low, virtually all stocks ended up with returns near the mean, so investment advice was not as valuable. (a) Suppose you want to design a study to determine whether the proportion of fund managers who outperform the market in low-dispersion years is less than the proportion of fund managers who outperform the market in high-dispersion years. What would be the response variable in this study? What is the explanatory variable in this study? (b) What or who are the individuals in this study? (c) To what population does this study apply? (d) What would be the null and alternative hypothesis? (e) Suppose this study was conducted and the data yielded a \(P\) -value of \(0.083 .\) Explain what this result suggests.

Step by step solution

01

- Identify the Response Variable

The response variable is the main outcome of interest in the study. Here, it would be whether a fund manager outperforms the market or not. Therefore, the response variable is 'performance of fund managers'.

02

- Determine the Explanatory Variable

The explanatory variable is the variable that explains changes in the response variable. In this case, it is whether the year is a low-dispersion year or a high-dispersion year. Thus, the explanatory variable is 'market dispersion'.

03

- Identify the Individuals in the Study

The individuals in the study are the entities being observed or analyzed. Here, the individuals are 'stock fund managers'.

04

- Determine the Population

The population in a study is the entire group about which information is being gathered. The study applies to 'all stock fund managers'.

05

- State Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) represents the default or no-effect scenario. The alternative hypothesis (\(H_a\)) represents the research hypothesis. Here:\(H_0\): The proportion of fund managers who outperform the market is the same in low-dispersion years and high-dispersion years.\(H_a\): The proportion of fund managers who outperform the market in low-dispersion years is less than in high-dispersion years.

06

- Analyze the P-value

The P-value is used to determine the significance of the results. A P-value of 0.083 indicates that there is an 8.3% probability of obtaining a result at least as extreme as the observed result, assuming the null hypothesis is true. Since 0.083 is greater than the common significance level of 0.05, we do not have enough evidence to reject the null hypothesis.

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Key Concepts

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

Response Variable

In any study, the response variable is the main outcome we are interested in measuring or observing. It's sometimes called the dependent variable. In the context of this exercise, the response variable is the 'performance of fund managers'. We are specifically looking at whether fund managers outperform the market or not. This is crucial because it directly answers the study's research question about the effectiveness of fund managers in different market conditions.

Explanatory Variable

An explanatory variable, also known as the independent variable, is the one that we believe might explain changes in the response variable. In our study, the explanatory variable is 'market dispersion'. This variable categorizes years into low-dispersion and high-dispersion based on how spread out the stock returns are. By assessing how this dispersion impacts fund manager performance, we can explore relationships between market conditions and investment success.

Null Hypothesis

The null hypothesis (\(H_0\)) is a statement assuming no effect or no difference. It serves as the starting point in hypothesis testing. In our case:

• \texttt{\textbf{\boldsymbol{H_0:}}} The proportion of fund managers who outperform the market is the same in low-dispersion years and high-dispersion years.

This hypothesis suggests that market conditions (low or high dispersion) do not affect the fund managers' ability to outperform the market. It's the scenario we assume to be true unless the data provides strong enough evidence to reject it.

Alternative Hypothesis

The alternative hypothesis (\(H_a\)) represents the research hypothesis, which states there is an actual effect or difference. For our study:

• \texttt{\textbf{\boldsymbol{H_a:}}} The proportion of fund managers who outperform the market in low-dispersion years is less than in high-dispersion years.

This hypothesis posits that market conditions significantly impact fund manager performance, with fewer managers outperforming during low-dispersion years. If the data supports this hypothesis, it would indicate that fund managers are more effective in high-dispersion years.

P-value

The P-value tells us the probability of obtaining a result at least as extreme as the one observed, assuming the null hypothesis is true. In our study, the P-value is 0.083. This means there is an 8.3% chance that the observed difference (or more extreme) in fund manager performance between low and high-dispersion years could occur if the null hypothesis were true. Since 0.083 is greater than the common significance level of 0.05, we don't have enough evidence to reject the null hypothesis. Therefore, we conclude that the data does not show a statistically significant difference in fund manager performance based on market dispersion.