Step 1 :The problem is asking us to perform a hypothesis test for the standard deviation. The null hypothesis is that the standard deviation is less than or equal to 3 grams, and the alternative hypothesis is that the standard deviation is more than 3 grams.
Step 2 :We are given a sample size of 24, a sample standard deviation of 3.4 grams, and a significance level of 0.05.
Step 3 :We need to calculate the test statistic and compare it to the critical value to decide whether to reject or fail to reject the null hypothesis.
Step 4 :Given that the sample size (n) is 24, the sample standard deviation (s) is 3.4 grams, the population standard deviation (sigma) is 3 grams, and the significance level (alpha) is 0.05.
Step 5 :The chi-square value is calculated as \(\chi^2 = \frac{(n-1)s^2}{\sigma^2} = \frac{(24-1)3.4^2}{3^2} = 29.54\)
Step 6 :The critical value for a chi-square distribution with n-1 degrees of freedom at a significance level of 0.05 is 35.17.
Step 7 :Since the chi-square value is less than the critical value, we fail to reject the null hypothesis.
Step 8 :Thus, there is insufficient evidence at a 0.05 level of significance that the bags should fail inspection.
Step 9 :\(\boxed{\text{Final Answer: We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.05 level of significance that the bags should fail inspection.}}\)