Final answer:
To test the hypothesis, we can use the Chi-Square test for variance. The calculated test statistic falls between the critical values, so we fail to reject the null hypothesis. There is not enough evidence to conclude that the variance of the bottle refilling process is different from 20 mg.
Step-by-step explanation:
To test the hypothesis, we can use the Chi-Square test for variance.
Step 1: State the null and alternative hypotheses.
Null hypothesis (H0): The variance of the bottle refilling process is 20 mg.
Alternative hypothesis (Ha): The variance of the bottle refilling process is not equal to 20 mg.
Step 2: Set the significance level (α). Let's assume α = 0.05.
Step 3: Compute the test statistic.
The test statistic is given by χ2 = (n-1) * s^2 / σ^2, where n is the sample size, s is the sample standard deviation, and σ is the hypothesized variance.
χ2 = (12-1) * (4.8)^2 / 20 = 13.824
Step 4: Compare the test statistic with the critical value from the Chi-Square distribution table.
Since this is a two-tailed test, the critical values are χ2(α/2, n-1) and χ2(1-α/2, n-1), where α/2 = 0.025 for α = 0.05 and n-1 = 11.
The critical values are χ2(0.025, 11) = 19.68 and χ2(0.975, 11) = 2.70.
Step 5: Make a decision.
If the test statistic is less than the lower critical value or greater than the upper critical value, we reject the null hypothesis.
In this case, the test statistic (13.824) falls between the critical values (2.70 and 19.68). Therefore, we fail to reject the null hypothesis. There is not enough evidence to conclude that the variance of the bottle refilling process is different from 20 mg.