Question: A study is done to determine which of two soft drinks has more sugar. There are 23 cans of Beverage…

A study is done to determine which of two soft drinks has more sugar. There are 23 cans of Beverage A in the researchers’ sample and 18 cans of Beverage B. The mean amount of sugar in Beverage A is 32.5 grams per can with a sample standard deviation of 0.8 grams. The mean amount of sugar in Beverage B is 31.9 grams with a sample standard deviation of 0.4 grams. The researchers believe that Beverage A has less sugar than Beverage B, on average. Both populations are normally distributed. Conduct a hypothesis test using a 2% level of significance.

Solution

To conduct a hypothesis test comparing the mean sugar content of Beverage A and Beverage B, we will follow these steps: \[ \begin{cases} \text{Step 1: State the hypotheses} & \\ \text{Step 2: Calculate the standard error} & \\ \text{Step 3: Determine the test statistic} & \\ \text{Step 4: Find the critical value} & \\ \text{Step 5: Make a decision} & \\ \text{Step 6: State the conclusion} & \end{cases} \] \[ \text{Step 1: State the hypotheses} \] \[ H_0: \mu_A = \mu_B & \] \[ H_1: \mu_A \neq \mu_B & \] \[ \text{Step 2: Calculate the standard error} \] \[ SE = \sqrt{\left(\frac{s_A^2}{n_A}\right) + \left(\frac{s_B^2}{n_B}\right)} = \sqrt{\left(\frac{0.8^2}{23}\right) + \left(\frac{0.4^2}{18}\right)} = \sqrt{\left(\frac{0.64}{23}\right) + \left(\frac{0.16}{18}\right)} = \sqrt{0.0278 + 0.0089} = \sqrt{0.0367} \approx 0.191 & \] \[ \text{Step 3: Determine the test statistic} \] \[ z = \frac{\bar{x}_A - \bar{x}B}{SE} = \frac{32.5 - 31.9}{0.191} \approx \frac{0.6}{0.191} \approx 3.14 & \] \[ \text{Step 4: Find the critical value} \] \[ \text{For a two-tailed test at } \alpha = 0.02, \text{ the critical z-values are } \pm z{\alpha/2} = \pm 2.326 & \] \[ \text{Step 5: Make a decision} \] \[ |z| = 3.14 > 2.326 \implies \text{Reject } H_0 & \] \[ \text{Step 6: State the conclusion} \] \[ \text{There is sufficient evidence at the } 2\% \text{ significance level to conclude that the mean sugar content of Beverage A is different from that of Beverage B.} & \]