## Example 64 Confidence Interval for the Population Mean of Sharpe Ratiosz Statistic

Suppose an investment analyst takes a random sample of U.S. equity mutual funds and calculates the average Sharpe ratio. The sample size is 100, and the average Sharpe ratio is 0.45. The sample has a standard deviation of 0.30. Calculate and interpret the 90 percent confidence interval for the population mean of all U.S. equity mutual funds by using a reliability factor based on the standard normal distribution.

The reliability factor for a 90 percent confidence interval, as given earlier, is Z0.05 = 1-65. The confidence interval will be

X±Z005 -Jf = 0.45 ± 1.65 —Jzrr = 0.45 ± 1.65(0.03) = 0.45 ± 0.0495 Vn V100

The confidence interval spans 0.4005 to 0.4995, or 0.40 to 0.50, carrying two decimal places. The analyst can say with 90 percent confidence that the interval includes

JhfijMjpulationjiiean._

In this example, the analyst makes no specific assumption about the probability distribution describing the population. Rather, the analyst relies on the central limit theorem to produce an approximate normal distribution for the sample mean.

As Example 6-4 shows, even if we are unsure of the underlying population distribution, we can still construct confidence intervals for the population mean as long as the sample size is large because we can apply the central limit theorem.

We now turn to the conservative alternative, using the ¿-distribution, for constructing confidence intervals for the population mean when the population variance is not known. For confidence intervals based on samples from normally distributed populations with unknown variance, the theoretically correct-reliability factor is based on the i-distribution. Using a reliability factor based on the i-distribution is essential for a small sample size. Using a i reliability factor is appropriate when the population variance is unknawn. even when we have a large sample and could use the central limit theorem to justify using a z reliability factor. In this large sample case, the i-distribution provides more-conservative (wider) confidence intervals.

The i-distribution is a symmetrical probability distribution defined by a single parameter known as degrees of freedom (df). Each value for the number of degrees of freedom defines one distribution in this family of distributions. We will shortly compare i-distributions with the standard normal distribution, but first we need to understand the concept of degrees of freedom. We can do so by examining the calculation of the sample variance.

Equation 6-3 gives the unbiased estimator of the sample variance that we use. The term in the denominator, n - 1, which is the sample size minus 1. is the number of degrees.

n - 1 as the number of degrees of freedom for determining reliability factors based on the i-distribution. The term "degrees of freedom" is used because in a random sample, we assume that observations are selected independently of each other. The numerator of the sample variance, however, uses the sample mean. How does the use of the sample mean affect the number of observations collected independently for the sample variance formula? With a sample of size 10 and a mean of 10 percent, for example, we can freely select only 9 observations. Regardless of the 9 observations selected, we can always find the value for the 10th observation that gives a mean equal to 10 percent. From the standpoint of the sample variance formula, then, there are 9 degrees of freedom. Given that we must, first compute the sample mean from the total of n independent observations, only n - 1 observations can be chosen independently for the calculation of the sample variance. The concept of degrees of freedom comes up frequently in statistics, and you will see it often in later chapters. _

f -j Suppose we sample from a normal distribution. The ratio z = {X - p,)/(a/Vn) is dis tributed normally with a mean of 0 and standard deviation of 1; however, the ratio t = (X - \x)l{sl\fn) follows the /-distribution with a mean of 0 and n - 1 degrees of free-' dom. The ratio represented by t is not normal because t is the ratio of two random vari-

¿Si ables, the sample mean and the sample standard deviation. The definition of the standard hJ(o. A.-;) normal random variable involves only one random variable, the sample mean. As degrees t s * ~M of freedom increase, however, the f-distribution approaches the standard normal distribu-

tion. Figure 6-1 shows the standard normal distribution and two i-distributions, one with ^ df = 2 and one with df = 8.

## Lessons From The Intelligent Investor

If you're like a lot of people watching the recession unfold, you have likely started to look at your finances under a microscope. Perhaps you have started saving the annual savings rate by people has started to recover a bit.

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