Questions: Location is known to affect the number, of a particular item, sold by Walmart. Two different locations, A and B, are selected on an experimental basis. Location A was observed for 18 days and location B was observed for 18 days. The number of the particular items sold per day was recorded for each location. On average, location A sold 39 of these items with a sample standard deviation of 9 and location B sold 55 of these items with a sample standard deviation of 6. Select a 99% confidence interval for the difference in the true means of items sold at location A and B. - [-22.95, -9.05] - [48.42, 61.58] - [87.42, 100.6] - [-18.55, -13.45] - [32.42, 45.58] - None of the above

Location is known to affect the number, of a particular item, sold by Walmart. Two different locations, A and B, are selected on an experimental basis. Location A was observed for 18 days and location B was observed for 18 days. The number of the particular items sold per day was recorded for each location. On average, location A sold 39 of these items with a sample standard deviation of 9 and location B sold 55 of these items with a sample standard deviation of 6. Select a 99% confidence interval for the difference in the true means of items sold at location A and B.
- [-22.95, -9.05]
- [48.42, 61.58]
- [87.42, 100.6]
- [-18.55, -13.45]
- [32.42, 45.58]
- None of the above

Transcript text: Location is known to affect the number, of a particular item, sold by Walmart. Two different locations, A and B , are selected on an experimental basis. Location A was observed for 18 days and location B was observed for 18 days. The number of the particular items sold per day was recorded for each location. On average, location A sold 39 of these items with a sample standard deviation of 9 and location B sold 55 of these items with a sample standard deviation of 6 . Select a $99 \%$ confidence interval for the difference in the true means of items sold at location A and B. [-22.95, -9.05] [48.42, 61.58] [87.42, 100.6] [-18.55, -13.45] [32.42, 45.58] None of the above

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with finding the 99% confidence interval for the difference in the true means of items sold at two different locations, A and B. The data provided includes:

Location A:
- Sample mean ($\bar{x}_1$) = 39
- Sample standard deviation ($s_1$) = 9
- Sample size ($n_1$) = 18
Location B:
- Sample mean ($\bar{x}_2$) = 55
- Sample standard deviation ($s_2$) = 6
- Sample size ($n_2$) = 18

Step 2: Calculate the Difference in Means

The difference in sample means is calculated as: \[ \bar{x}_1 - \bar{x}_2 = 39 - 55 = -16 \]

Step 3: Calculate the Standard Error

The standard error (SE) for the difference in means is calculated using the formula: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{9^2}{18} + \frac{6^2}{18}} = \sqrt{\frac{81}{18} + \frac{36}{18}} = \sqrt{4.5 + 2} = \sqrt{6.5} \]

Step 4: Determine the Critical Value

For a 99% confidence level, we find the critical value ($t$) from the t-distribution with degrees of freedom calculated using the formula for unequal variances. The critical value for this confidence level is approximately $t \approx 2.75$.

Step 5: Calculate the Margin of Error

The margin of error (ME) is calculated as: \[ ME = t \cdot SE = 2.75 \cdot \sqrt{6.5} \]

Step 6: Construct the Confidence Interval

The confidence interval for the difference in means is given by: \[ (\bar{x}_1 - \bar{x}_2) \pm ME = -16 \pm 2.75 \cdot \sqrt{6.5} \] Calculating the bounds: \[ \text{Lower bound} = -16 - 2.75 \cdot \sqrt{6.5} \approx -23.02 \] \[ \text{Upper bound} = -16 + 2.75 \cdot \sqrt{6.5} \approx -8.98 \]