An investor wants to invest his money in a fund which has maintained a steady value. A fund manager claims that one of his bond funds has maintained an average price of $\$ 7.00$ with a variance of 0.2 . In order to find out if the fund manager's claim is true, the investor samples the prices from 13 random days and finds a standard deviation of 0.2815 in the price. Can the investor conclude that the variance of the share price of the bond fund is less than claimed at $\alpha=0.1$ ? Assume the population is normally distributed.

Step 2 of 5: Determine the critical value(s) of the test statistic. If the test is two-tailed, separate the values with a comma. Round your answer to three decimal places.
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Step 1 ：The problem is asking for the critical value of the test statistic for a chi-square test. This test is used to test hypotheses about the variance or standard deviation of a population. In this case, we are testing the claim that the variance of the share price of the bond fund is less than 0.2.

Step 2 ：The critical value for a chi-square test is determined by the degrees of freedom and the level of significance. The degrees of freedom for a chi-square test about variance or standard deviation is n-1, where n is the sample size. In this case, the sample size is 13, so the degrees of freedom is 12.

Step 3 ：The level of significance, \(\alpha\), is given as 0.1. Since this is a one-tailed test (we are testing if the variance is less than a certain value), we will use the entire 0.1 as the level of significance.

Step 4 ：We can find the critical value from a chi-square distribution table, or we can use a statistical calculator or software to find it. In this case, the critical value is approximately 18.549.

Step 5 ：Final Answer: The critical value of the test statistic for a chi-square test with 12 degrees of freedom and a level of significance of 0.1 is approximately \(\boxed{18.549}\).