Final answer:
The question involves using a chi-square test to assess the significance of the increase in variability of the weights of the packages. By comparing the calculated chi-square value to the critical value at a specific level of significance, we determine if the increase is statistically significant.
Step-by-step explanation:
The question asks whether the apparent increase in the variability of the weights of certain packages is significant. To address this, a statistical hypothesis test for variance known as the chi-square test can be used.
The null hypothesis (H0) is that the machine's variability has not increased (i.e., the population variance is equal to the square of the previous standard deviation, which is 7.1 grams), and the alternative hypothesis (H1) is that the machine's variability has increased, as suggested by the sample's larger standard deviation of 9.1 grams.
To perform the test, we calculate the chi-square test statistic using the formula Chi-Square (X2) = (n-1)S2 / σ2, where n is the sample size, S is the sample standard deviation, and σ2 is the population variance.
For the sample of 20 packages, with S = 9.1 grams, n = 20, and the historical standard deviation of 7.1 grams (so σ2 = 7.12), we would then compare the calculated chi-square value to the critical chi-square value from the chi-square distribution table at the desired level of significance to determine if the increase in variability is statistically significant.