[DATA] Filling BottlesA certain brand of apple juice is supposed to have 64 ounces of juice. Because the penalty for underfilling bottles is severe, the target mean amount of juice is 64.05 ounces. However, the filling machine is not precise, and the exact amount of juice varies from bottle to bottle. The quality-control manager wishes to verify that the mean amount of juice in each bottle is 64.05 ounces so that she can be sure that the machine is not over- or underfilling. She randomly samples 22 bottles of juice, measures the content, and obtains the following data:


A normal probability plot suggests the data could come from a population that is normally distributed. A boxplot does not show any outliers.



a. Should the assembly line be shut down so that the machine can be recalibrated? Use a 0.01 level of significance.

Verified step by step guidance

1. Step 1: Define the hypotheses for the test. The null hypothesis \(H_0\) is that the mean amount of juice per bottle is equal to the target mean, i.e., \(\mu = 64.05\) ounces. The alternative hypothesis \(H_a\) is that the mean is not equal to 64.05 ounces, i.e., \(\mu \neq 64.05\) ounces. This is a two-tailed test.
2. Step 2: Calculate the sample mean \(\bar{x}\) and the sample standard deviation \(s\) from the 22 bottle measurements provided. Use the formulas: \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\) \(s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\) where \(n=22\) is the sample size.
3. Step 3: Compute the test statistic \(t\) using the formula: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\) where \(\mu_0 = 64.05\) ounces is the hypothesized population mean.
4. Step 4: Determine the critical value(s) for a two-tailed t-test at the \(\alpha = 0.01\) significance level with \(df = n - 1 = 21\) degrees of freedom. You can find this value from a t-distribution table or statistical software.
5. Step 5: Compare the calculated test statistic \(t\) to the critical value(s). If \(|t|\) is greater than the critical value, reject the null hypothesis \(H_0\) and conclude that the machine should be recalibrated. Otherwise, do not reject \(H_0\) and conclude there is not enough evidence to shut down the assembly line.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence to reject a null hypothesis. In this context, the null hypothesis states that the mean amount of juice is 64.05 ounces. The alternative hypothesis would suggest the mean differs from this target, guiding decisions about recalibrating the machine.

Significance Level and p-value

The significance level (α) is the threshold for rejecting the null hypothesis, set here at 0.01. It represents a 1% risk of concluding the mean differs when it actually does not. The p-value measures the probability of observing the sample data if the null hypothesis is true; if the p-value is less than α, the null hypothesis is rejected.

Normality Assumption and Sample Size

Many hypothesis tests assume the data come from a normally distributed population, especially with small samples. The normal probability plot and absence of outliers support this assumption here. With a sample size of 22, the Central Limit Theorem partially applies, but verifying normality ensures the validity of the t-test used for inference.