9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Variance

9. Hypothesis Testing for One Sample

Performing Hypothesis Tests: Variance

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Multiple Choice

A machine produces ball bearings that are designed to have a diameter standard deviation of 0.04 mm, but an engineer suspects the variability has increased. A sample of 60 bearings shows a standard deviation of 0.052 mm. Perform a hypothesis test with $α = 0.01$ to test the claim. Should the line manager have the machine serviced?

Since $P$ -value $=7.59\times10^{-4}<0.01=\alpha$ , we fail to reject $H_0$ , not enough evidence to suggest $\sigma>0.04$ .

No need to service the machine.

Since $P$ -value $= 0.0016 < 0.01 = α$ , we fail to reject $H_0$ , not enough evidence to suggest $\sigma>0.04$ .

No need to service the machine.

Since $P$ -value $= 7.59 \times 10^{- 4} < 0.01 = α$ , we reject $H_{0}$ , enough evidence to suggest $σ > 0.04$ .

Machine should be serviced.

Since $P$ -value $=0.0016<0.01=\alpha$ , we reject $H_0$ , enough evidence to suggest $\sigma>0.04$ .

Machine should be serviced.

Verified step by step guidance

Step 1: Define the hypotheses for the test. Since the engineer suspects the variability has increased, the null hypothesis and alternative hypothesis are: \[ H_0: \sigma = 0.04 \] \[ H_a: \sigma > 0.04 \] This is a right-tailed test for the population standard deviation.

Step 2: Identify the significance level and sample information. Here, the significance level is given as: \[ \alpha = 0.01 \] The sample size is: \[ n = 60 \] The sample standard deviation is: \[ s = 0.052 \]

Step 3: Calculate the test statistic using the chi-square distribution formula for variance: \[ \chi^2 = \frac{(n-1)s^2}{\sigma_0^2} \] where $\sigma_0 = 0.04$ is the hypothesized population standard deviation under $H_0$.

Step 4: Determine the critical value or p-value. Since this is a right-tailed test, find the critical chi-square value for $\alpha = 0.01$ with degrees of freedom: \[ df = n - 1 = 59 \] Alternatively, calculate the p-value corresponding to the test statistic from Step 3 using the chi-square distribution with 59 degrees of freedom.

Step 5: Make a decision by comparing the p-value to the significance level $\alpha$: - If $p$-value $< \alpha$, reject $H_0$ and conclude there is enough evidence to suggest the variability has increased. - If $p$-value $\geq \alpha$, fail to reject $H_0$ and conclude there is not enough evidence to suggest an increase in variability. Based on this decision, advise whether the machine should be serviced.

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