91Ó°ÊÓ

Annual holdings turnover for a mutual fund is the percentage of a fund's assets that are sold during a particular year. Generally speaking, a fund with a low value of turnover is more stable and risk averse, whereas a high value of turnover indicates a substantial amount of buying and selling in an attempt to take advantage of short-term market fluctuations. Here are values of turnover for a sample of 20 large-cap blended funds (refer to Exercise \(1.53\) for a bit more information) extracted from Morningstar.com: \(\begin{array}{llllllllll}1.03 & 1.23 & 1.10 & 1.64 & 1.30 & 1.27 & 1.25 & 0.78 & 1.05 & 0.64 \\ 0.94 & 2.86 & 1.05 & 0.75 & 0.09 & 0.79 & 1.61 & 1.26 & 0.93 & 0.84\end{array}\) a. Would you use the one-sample \(t\) test to decide whether there is compelling evidence for concluding that the population mean turnover is less than \(100 \%\) ? Explain. b. A normal probability plot of the \(20 \ln\) (turnover) values shows a very pronounced linear pattern, suggesting it is reasonable to assume that the turnover distribution is lognormal. Recall that \(X\) has a lognormal distribution if \(\ln (X)\) is normally distributed with mean value \(\mu\) and variance \(\sigma^{2}\). Because \(\mu\) is also the median of the \(\ln (X)\) distribution, \(e^{\mu}\) is the median of the \(X\) distribution. Use this information to decide whether there is compelling evidence for concluding that the median of the turnover population distribution is less than \(100 \%\).

Step by step solution

01

Check Conditions for Using One-Sample t-Test

To use a one-sample t-test, the sample data should approximately follow a normal distribution or be sufficiently large (usually n > 30). Here, the sample size is 20, which is not large, so the distribution should ideally be normal. The problem mentions that a normal probability plot suggests a linear pattern for ln(turnover), indicating that ln(turnover) is approximately normal.

02

Analyze Given Data for Part A

The one-sample t-test is used to test if the population mean is a certain value. In part a, we are asked if the mean turnover is less than 100%, but the turnover values are already expressed as more than 100% (e.g., 1.03 represents 103%). Since the mean value must be tested against a hypothesized value (less than 1.0 for this problem), it seems inappropriate to apply a one-sample t-test directly due to distribution assumptions.

03

Deciding Test Suitability for Part A

Since the assumption of normality is not met for original turnover values but is met for the logged values, using the one-sample t-test directly on the original turnover values might not be applicable. ln(turnover) is normal, so considering transformations might help if strictly needed, but as of part a, the data do not well support direct population mean hypothesis testing.

04

Analyze Log-Transformed Data for Part B

Given ln(turnover) has a normal distribution, it's appropriate to test the median of the original turnover distribution by considering ln(turnover). The median of a lognormal distribution is e^μ where μ is the mean of ln(turnover). Since the test hypothesis is that the median (e^μ) is less than 1 (100%), a one-sample t-test on ln(turnover) can be applied to check if μ < 0.

05

Calculate Sample Statistics

Compute the sample mean and standard deviation of the ln(turnover) values. Use these values to perform the t-test: - mean = sample mean- s = sample standard deviation- n = sample size (20)Then calculate the t-statistic: \[ t = \frac{\text{mean} - 0}{s/\sqrt{n}} \]

06

Perform t-Test and Interpret Results for Part B

Using the t-statistic and degrees of freedom (n-1 = 19), determine the p-value for a one-tailed test. If the p-value is less than the significance level (commonly 0.05), reject the null hypothesis and conclude that there's compelling evidence that the median turnover is less than 100%.

07

Final Conclusion

After calculating, if you find that the p-value is small enough, conclude that there's sufficient evidence that the median turnover distribution is less than 100% (e^0).

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Understanding Lognormal Distribution

The concept of a lognormal distribution is related to the transformation of data. This means that a variable is considered to follow a lognormal distribution if its natural logarithm is normally distributed.
This is essential in many fields, especially in finance and environmental studies, because many real-world data sets tend to show lognormal characteristics.

• If you take the natural log (\(\ln(X)\)) of the data set, and it plots as a linear line on a normal probability plot, then the original data (\(X\)) exhibits a lognormal distribution.
• This means that values are skewed positively (to the right), indicating there are a few high values pulling the "tail" out.
• In a lognormal distribution, all the values are positive, which makes it crucial for modeling financial data, like stock prices or mutual fund turnovers.

Understanding this helps you analyze data with skewed characteristics and allows for the application of statistical methods such as the t-test with log-transformed values.

The Significance of Median in Data Analysis

The median is a vital measure in statistical analysis, often used alongside or even in place of the mean. It represents the middle value in a dataset when values are arranged in ascending order. Here's why the median is crucial:

• Unlike the mean, the median is less affected by outliers and skewed data, which ensures a more accurate representation of the central tendency for skewed distributions.
• In a lognormal distribution, while \(\ln(X)\) is normally distributed, \( e^{\mu}\) (where \(\mu\) is the mean of \(\ln(X)\)), is the median of the original lognormal distribution.
• Thus, finding the median can be particularly useful in non-normal data contexts, like our exercise with turnover rates, where it could give a more realistic expectation behind the metrics.

This emphasis on the median allows us to make judgments about the data's behavior, especially under transformations like the logarithm.

How to Use a Normal Probability Plot

A normal probability plot is a graphical tool used to assess if a dataset follows a normal distribution. This plot is incredibly useful because linearity in the plot suggests normality in the data.

• To create one, you graph the ordered data values against the expected quantiles of a normal distribution.
• If the plot points roughly form a straight line, it indicates that the data is approximately normally distributed.
• In this exercise, the normal probability plot of the \(\ln(turnover)\) demonstrates that the log-transformed data is normally distributed.

Such a visual inspection is essential before applying certain statistical tests, as these tests often rely on the assumption of normal distribution in the dataset.

The Basics of Population Mean Hypothesis Testing

Population mean hypothesis testing allows you to make inferences about a population based on sample data. It involves comparing the sample data against a null hypothesis, which is a statement of no effect or no difference. Here are the steps involved:

• State your null and alternative hypotheses. In our exercise, the null hypothesis might be that the population median turnover is equal to 100%.
• Use a statistical test, such as the t-test for means, to calculate the test statistic from your sample data. This involves using formulas such as \( t = \frac{\text{sample mean} - \text{hypothesized value}}{\text{standard deviation}/\sqrt{n}} \).
• Calculate the p-value, which helps determine the strength of evidence against the null hypothesis. A p-value less than the alpha level (e.g., 0.05) indicates strong evidence against the null hypothesis.
• Finally, draw a conclusion whether to reject the null hypothesis or not based on the p-value.

This process was applied to the log-transformed turnover rates in part B of our exercise, leading to conclusions about the median turnover.