Level 1 CFA® Exam:
Tests Concerning Variance
CFA Exam: Tests Concerning Single Variance of Normally Distributed Population
star content check off when doneLet \(\sigma^{2}_{0}\) be the hypothesized value of variance.
In tests concerning a single variance of a normally distributed population, we make use of a chi-square \(\chi^2\) test statistic. A \(\chi^2\) distribution:
- is defined by one parameter, namely the number of degrees of freedom,
- is asymmetrical, and
- is bounded below by zero.
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So, rejection points for hypothesis tests are as follows:
- For a two-sided hypothesis test: we reject the null hypothesis if the test statistic is greater than the value for the chi-square distribution with a probability of \(\frac{\alpha}{2}\) in the right tail OR less than the value for the \(\chi^2\) distribution with a probability of \(\frac{\alpha}{2}\) in the left tail. Assuming what we said before, in the table for probability in right tail we check if the value of the test is greater than the value for the \(\chi^2\) distribution with a probability of \(\frac{\alpha}{2}\) in the right tail or less than the value for the \(\chi^2\) distribution with a probability of \(1-\frac{\alpha}{2}\) in the right tail.
- For a right-tailed test: we reject the null hypothesis if the test statistic is greater than the value for the \(\chi^2\) distribution with a probability of \(\alpha\) in the right tail.
- For a left-tailed test: we reject the null hypothesis if the test statistic is less than the value for the \(\chi^2\) distribution with a probability of \(\alpha\) in the left tail or with a probability of \(1-\alpha\) in the right tail.
Glenn company is producing steel rods using a special production machine. The primary product the company sells is a steel rod with diameter equal to 24 mm. Company’s customers pay much attention to possible deviations from the mean diameter – they accept rods with variance less than 6.25 mm squared. John Clark, chief of machine operators, decided to test whether the machine needs a new setup. He picked up 27 rods and measured their diameter and calculate the variance. According to his findings, the variance is equal to 7.29 mm squared. Help John test hypothesis regarding the machine’s new setup. Assume 5% level of significance and a normally distributed population.
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CFA Exam: Tests Concerning Equality or Inequality of Two Variances of Normally Distributed Populations
star content check off when doneNow let’s have a look at tests concerning the equality or inequality of two variances of normally distributed populations.
Assuming that the samples drew from these two populations are independent, we can test them with an F-test. The F-test is the ratio of sample variances.
This test concerning the equality or inequality of variances of two populations makes use of the F-distribution. Like the chi-square distribution, the F-distribution is a family of asymmetrical distributions bounded from below by 0. Each F-distribution is defined by two values of degrees of freedom, called the numerator and denominator degrees of freedom.
While conducting the F-test, we have to compute the sample variances for both populations. Note that we put the greater variance in the numerator and the lower one in the denominator of the formula. So the value of the F-test is always greater than or equal to zero.
Rejection points for hypothesis tests are as follows:
- For a two-sided hypothesis test: we reject the null hypothesis if the test statistic is greater than the value for the F-distribution with a significance level of \(\frac{\alpha}{2}\) and the numerator and denominator degrees of freedom given.
- For a right- and left-tailed test: we reject the null hypothesis if the test statistic is greater than the value for the F-distribution with a significance level of \(\alpha\) and the numerator and denominator degrees of freedom given.
Glenn company is one of the main companies in the UKA district producing steel rods. The managers of Glenn company decided to purchase a new production machine to increase production and sales. Fine Machinery, the company that sells steel rod production machines, offer Glenn company two production machines: Great Machine and CWFW Machine. Mike Stand, the sales manager of Fine Machinery, explains Glenn managers that 'CWFW' stands for "complete and without faults or weaknesses". Mike Stand argues that the variance of diameter of a steel rod manufactured using CFWF Machine is much lower than using Great Machine. Glenn managers decide to test Great Machine and CFWF Machine. The table presents their findings:
| sample variance | sample size | |
|---|---|---|
| Great Machine | 5.76 | 22 |
| CFWF Machine | 2.56 | 19 |
Help Glenn managers decide whether with 5% confidence level CFWF Machine manufactures steel rods with a smaller variance of diameter. Assume the normal distribution for both populations.
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- In tests concerning a single variance of a normally distributed population, we make use of a chi-square test statistic.
- A chi-square distribution is defined by one parameter (the number of degrees of freedom), is asymmetrical, and is bounded below by zero.
- In tests concerning the equality or inequality of two variances of normally distributed populations, assuming that the samples drew from these two populations are independent, we can use an F-test. The F-test is the ratio of sample variances.
- Each F-distribution is defined by two values of degrees of freedom, called the numerator and denominator degrees of freedom.