(a) What is the 95% confidence interval for the standard deviation of the number of carbohydrates in processed deli meats? Does this interval suggest that the manufacturer's claim is inaccurate?
(b) Can we determine if the dietitian has evidence to support the claim that the standard deviation of carbohydrates in deli meats is higher than what the manufacturer states?
(a) The 95% confidence interval for the standard deviation of carbohydrates in deli meats is 0.438 to 0.775 grams per serving. This interval provides evidence against the manufacturer's claim of a standard deviation of 0.5 grams per serving. The confidence interval calculation suggests that the true standard deviation may differ from the reported value.
(b) The dietitian does have evidence that the standard deviation of carbohydrates in deli meats is higher than the manufacturer's claim.
Confidence Interval Analysis
The 95% confidence interval for the standard deviation of carbohydrates in deli meats is (0.438, 0.775) grams per serving. This interval does provide evidence against the manufacturer's claim of a standard deviation of 0.5 grams per serving. Since the confidence interval does not include the manufacturer's claimed value, it suggests that the true standard deviation of the number of carbohydrates in deli meats is likely different from the reported value.
To calculate the confidence interval, we use the formula:
Confidence Interval = (sample standard deviation) * sqrt((n-1)/χ²(α/2, n-1)) to (sample standard deviation) * sqrt((n-1)/χ²(1-α/2, n-1))
Where n is the sample size, α is the significance level, and χ² represents the chi-square distribution. In this case, with n = 18 and α = 0.05, the critical values from the chi-square distribution table for α/2 = 0.025 and 1-α/2 = 0.975 with 17 degrees of freedom were found to be approximately 29.707 and 7.564, respectively.
Plugging in the values, the confidence interval calculated is (0.438, 0.775) grams per serving. Since this interval does not contain the manufacturer's claimed value of 0.5 grams per serving, it provides evidence against the claim.
Significance Test Analysis
The dietitian does have evidence that the standard deviation of carbohydrates in deli meats is higher than what is advertised by the manufacturer. By applying a significance test using the chi-square distribution and calculating the test statistic, the evidence supports this conclusion.
The null hypothesis (H0) states that the standard deviation is equal to the manufacturer's claimed value of 0.5 grams per serving, while the alternative hypothesis (Ha) is that the standard deviation is higher. The test statistic is calculated to be 22.896, which is then compared to the critical value from the chi-square distribution at α = 0.05 and degrees of freedom (n - 1) = 17. The critical value was determined to be approximately 30.191.
Since the test statistic (22.896) is not greater than the critical value (30.191), the null hypothesis cannot be rejected. The computed p-value also supports this decision, as it is greater than 0.05. Therefore, the dietitian lacks sufficient evidence to conclude that the standard deviation is higher than the manufacturer's stated value.