Questions: A sample of 36 cans of regular Coke produced a mean weight of 0.8168 pounds. A sample of 36 cans of regular Pepsi produced a mean weight of 0.8241 pounds. Coca-Cola reports that standard deviation for the weight of all cans of regular Coke is known to be 0.0075 pounds and Pepsi-Co reports that the standard deviation for the weight of all cans of regular Pepsi is known to be 0.0057 pounds. Construct a 99% confidence interval for the difference between the mean weights of regular Coke and regular Pepsi.

Solution Steps

We have the following data for the two samples:

• For regular Coke:

  • Sample mean (\(\bar{x}_1\)) = 0.8168 pounds
  • Sample size (\(n_1\)) = 36
  • Population standard deviation (\(\sigma_1\)) = 0.0075 pounds

• For regular Pepsi:

  • Sample mean (\(\bar{x}_2\)) = 0.8241 pounds
  • Sample size (\(n_2\)) = 36
  • Population standard deviation (\(\sigma_2\)) = 0.0057 pounds

We are constructing a 99% confidence interval, which corresponds to a significance level (\(\alpha\)) of: \[ \alpha = 1 - 0.99 = 0.01 \]

The formula for the confidence interval for the difference between two population means with known variances is given by: \[ (\bar{x}_1 - \bar{x}_2) \pm z \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \] Where \(z\) is the z-score corresponding to the desired confidence level. For a 99% confidence level, \(z \approx 2.5758\).

Substituting the values: \[ \text{Difference of means} = \bar{x}_1 - \bar{x}_2 = 0.8168 - 0.8241 = -0.0073 \] \[ \text{Standard Error} = \sqrt{\frac{0.0075^2}{36} + \frac{0.0057^2}{36}} = \sqrt{\frac{0.00005625}{36} + \frac{0.00003249}{36}} = \sqrt{0.0000015625 + 0.0000009025} = \sqrt{0.000002465} \approx 0.00157 \]

Now, we can calculate the confidence interval: \[ -0.0073 \pm 2.5758 \cdot 0.00157 \] Calculating the margin of error: \[ \text{Margin of Error} \approx 2.5758 \cdot 0.00157 \approx 0.00405 \]

Thus, the confidence interval is: \[ (-0.0073 - 0.00405, -0.0073 + 0.00405) = (-0.01135, -0.00325) \]

The 99% confidence interval for the difference between the mean weights of regular Coke and regular Pepsi is: \[ (-0.0113, -0.0033) \]

Final Answer