Claim about a Question 7, 8.4.11-T HW Score: $28.89 \%, 4.33$ of 15 points Part 2 of 4 Points: 0 of 1 Save Workers at a certain soda drink factory collected data on the volumes (in ounces) of a simple random sample of 20 cans of the soda drink. Those volumes have a mean of $12.19 \mathrm{oz}$ and a standard deviation of $0.11 \mathrm{oz}$, and they appear to be from a normally distributed population. If the workers want the filling process to work so that almost all cans have volumes between $11.92 \mathrm{oz}$ and $12.48 \mathrm{oz}$, the range rule of thumb can be used to estimate that the standard deviation should be less than $0.14 \mathrm{oz}$. Use the sample data to test the claim that the population of volumes has a standard deviation less than $0.14 \mathrm{oz}$ Use a 0.025 significance level. Complete parts (a) through (d) below. a. Identify the null and alternative hypotheses. Choose the correct answer below. A \[ \begin{array}{l} H_{0}: \sigma=0.14 \text { oz } \\ H_{1}: \sigma<0.14 \text { oz } \end{array} \] C. \[ \begin{array}{l} H_{0}: \sigma>0.14 \mathrm{oz} \\ H_{1}: \sigma=0.14 \mathrm{oz} \end{array} \] B. \[ \begin{array}{l} H_{0}: \sigma \geq 0.14 \mathrm{oz} \\ H_{1}: \sigma<0.14 \mathrm{oz} \end{array} \] D. \[ \begin{array}{l} H_{0}: \sigma=0.14 \mathrm{oz} \\ H_{1}: \sigma \neq 0.14 \mathrm{oz} \end{array} \] b. Compute the test statistic. \[ x^{2}=\square \] (Round to three decimal places as needed.)
Solution
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis is typically a statement of no effect or status quo, and the alternative hypothesis is what we are testing against the null hypothesis. In this case, we are testing the claim that the population of volumes has a standard deviation less than $0.14 \mathrm{oz}$. Therefore, the null hypothesis should be that the standard deviation is not less than $0.14 \mathrm{oz}$, and the alternative hypothesis should be that the standard deviation is less than $0.14 \mathrm{oz}$. The null and alternative hypotheses are: \[ \begin{array}{l} H_{0}: \sigma \geq 0.14 \mathrm{oz} \ H_{1}: \sigma<0.14 \mathrm{oz} \end{array} \]
Step 2 ：Compute the test statistic. The test statistic for a hypothesis test regarding a population standard deviation is a chi-square statistic, which is calculated as $(n-1)s^2/\sigma^2$, where $n$ is the sample size, $s^2$ is the sample variance, and $\sigma^2$ is the population variance under the null hypothesis. In this case, $n=20$, $s=0.11 \mathrm{oz}$, and $\sigma=0.14 \mathrm{oz}$ under the null hypothesis.
Step 3 ：Calculate the chi-square statistic using the formula $(n-1)s^2/\sigma^2$. Substitute the given values into the formula: $chi_square = (20-1)*(0.11)^2/(0.14)^2$
Step 4 ：The test statistic is \(\boxed{11.730}\).
