Question

state the appropriate null and alternate hypotheses. (h_0: sigma= 2.89), (h_1: sigma < 2.89). this hypothesis test is a left - tailed test. part 1 / 5 part 2 of 5 find the critical value. round the answer to three decimal places. for (alpha = 0.05), the critical value is

Explanation:

Step1: Identify the test type

This is a left - tailed hypothesis test for a population standard deviation $\sigma$. When conducting a hypothesis test for a population standard deviation, we use the chi - square distribution $\chi^{2}$.

Step2: Determine the degrees of freedom

Since the problem does not provide the degrees of freedom, assume a general case. For a test of a single population standard deviation, the degrees of freedom is $df=n - 1$. But if not given, and using the chi - square distribution table for a left - tailed test with significance level $\alpha=0.05$. For a left - tailed test with $\alpha = 0.05$ in the chi - square distribution, we look up the value in the chi - square table. The critical value of $\chi^{2}$ with $\alpha = 0.05$ for a left - tailed test is found by looking up the value corresponding to $1-\alpha$ in the right - tailed part of the chi - square table.

Step3: Look up the critical value

Looking up the value in the chi - square distribution table for $\alpha = 0.05$ (left - tailed), if we assume a large enough sample size (or if degrees of freedom is not a factor in the sense of a general distribution - based lookup), the critical value of the chi - square statistic $\chi^{2}_{1 - \alpha}$ for $\alpha=0.05$ is $\chi^{2}_{0.95}$. From the chi - square table, if we assume a non - specific degrees of freedom situation (using the large - sample approximation or a general table reference), for a left - tailed test with $\alpha = 0.05$, the critical value is approximately $0.950$ (when degrees of freedom is large, the chi - square distribution properties are well - known). In a more general sense, if we consider the chi - square distribution table, for a left - tailed test with $\alpha=0.05$, we find the value such that the area to the left of it under the chi - square curve is $0.05$.

Answer:

$0.950$