Question

A sample of 16 students showed that the variance in the number of hours they spend studying is 23. At the 5% level of significance, test to see if the variance of the population is significantly different from 28. Use the critical value approach.

State the null and alternative hypotheses (in hours2). (Enter != for != as needed.)
H0:
Ha:
Find the value of the test statistic. (Round your answer to three decimal places.)

Answer 1

To test if the variance in study hours is significantly different from 28, a hypothesis test is conducted with H0: σ^2 = 28 and Ha: σ^2 != 28. The test statistic is calculated, and compared with chi-square distribution critical values at a 5% significance level.

Step-by-step explanation:

The question is about conducting a hypothesis test to determine whether the variance of the number of hours students spend studying is significantly different from a given value. We begin by stating the null hypothesis (H0) and the alternative hypothesis (Ha), both in terms of variance (hours2).

H0: σ2 = 28
Ha: σ2 != 28

The test statistic for variance is calculated using the formula χ2 = (n - 1)s2 / σ20, where n is the sample size, s2 is the sample variance, and σ20 is the variance under the null hypothesis. Substituting the given values, we get:

χ2 = (16 - 1)×23 / 28 ≈ 12.196

Using a chi-square distribution table and a 5% level of significance (α = 0.05), we find the critical values that correspond to the upper and lower tails of the distribution, given our degrees of freedom (df = 16 - 1 = 15). We compare our test statistic to these critical values to make a decision on whether to reject H0 or not.