Appropriate test statistic and hypothesis test concerning the mean difference of two normally distributed populations

CFA level I / Quantitative Methods: Application / Hypothesis Testing / Appropriate test statistic and hypothesis test concerning the mean difference of two normally distributed populations

When the samples from two populations are dependent, then we use a paired comparison test. The paired observations are observations that are dependent due to some common factor. For example, change in dividend policy before and after an acquisition by a company. Here, we take observations for the same company before and after an acquisition. The samples are dependent for this test. In this hypothesis test, we arrange the observations in pairs and take the difference between the paired observations. Let d_i denote the difference between two paired observations where d_i = x_Ai - x_Bi, where x_Ai and x_Bi are the ith pair of observations. If µ_d stands for the population mean difference and µ_d0is the hypothesized value for the population mean difference, then we can have the following possible hypotheses:

H₀: µ_{d =}µ_d0versus H_a: µ_{d ≠}µ_d0(two-tailed test)
H₀: µ_{d ≤}µ_d0versus H_a: µ_{d >}µ_d0(right-tailed test)
H₀: µ_{d ≥}µ_d0versus H_a: µ_{d <}µ_d0(left-tailed test)

The test statistic for the above hypotheses tests are given by t = (d ̅- µ_d0)/s_{d ̅}

where d ̅ is the mean of the difference between the paired observations (d_i). s_d is the standard deviation of the difference between the paired observations (d_i) and s_{d ̅} = s_d/√n where n is the sample size for the paired observations.

Degrees of freedom will be equal to n-1.

Example 5: Paired comparison test

You want to check whether the monthly mean return of mutual fund A is greater than or equal to the mean return of mutual fund B. The mutual fund A invests in the equity markets across the world whereas mutual fund B invests only in US equity markets. You take the data for 60 months from Jan 2011 to Dec 2015 and calculate the difference between the paired observations. The mean of the difference between the paired observations of monthly returns for mutual fund A and mutual fund B is 0.18 percent. The standard deviation of the difference between the paired observations is 2.40 percent. Formulate the null and alternative hypotheses and check the hypotheses testing at 1 percent level of significance.

Solution:

The hypotheses are given below:

H₀: µ_{d ≥}0 versus H_a: µ_{d <}0 (left-tailed test)

The degrees of freedom equals 59, and the test is a single-tailed test. The critical value of the test statistic is -2.39.

s_{d ̅}= s_d/√n = 0.024/√60 = 0.003098.

t-statistic = (d ̅- µ_d0)/s_{d ̅}= (0.0018 - 0)/0.003098 = 0.581.

The test statistic value lies in the null hypothesis region as 0.581 is greater than -2.39. So, we fail to reject the null hypothesis.

The mean monthly return of mutual fund A is statistically greater than or equal to the mean monthly return of mutual fund B at 1 percent level of significance.

Please note that we use paired comparison test here because the samples are dependent. The mutual fund A invests in both US and non-US equity markets. So, there is some overlap with the mutual fund B.

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