Question

The following data set provides information on wholesale sales by establishments and by total sales.

Number of establisments Total sales in $billions Product sales as a % of total sales
Wholesale 1 5378 273 11.6
Wholesale 2 4696 202 8.7
Wholesale 3 160 36 27.2
Merchants 1 4856 238 11.5
Merchants 2 4696 202 8.7
Merchants 3 160 36 27.2
Mean 3324.3 164.5 15.8
Standard deviation 2463.8 103 8.9

A cheese merchant is looking to expand her business. She looks at the data set about cheese establishments in six categories, in which the sample mean is 3,324.3 and the sample standard deviation is 2,463.8.

Find the lowest level of the 68% confidence interval estimate.

Answer 1

Explanation:

To find the confidence interval estimate for the mean, you can use the formula:

\[ \text{Confidence Interval} = \text{Mean} \pm \left( \text{Z-Score} \times \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \right) \]

For a 68% confidence interval, you typically use a Z-score of 1 (as 68% of the data falls within one standard deviation of the mean).

So, the formula becomes:

\[ \text{Confidence Interval} = \text{Mean} \pm \left( \frac{\text{Standard Deviation}}{\sqrt{\text{Sample Size}}} \right) \]

Using the provided data:

\[ \text{Confidence Interval} = 3324.3 \pm \left( \frac{2463.8}{\sqrt{1}} \right) \]

Now, calculate the margin of error:

\[ \text{Margin of Error} = \frac{2463.8}{\sqrt{1}} \]

\[ \text{Margin of Error} = 2463.8 \]

So, the 68% confidence interval estimate is:

\[ 3324.3 \pm 2463.8 \]

To find the lowest level of this confidence interval, subtract the margin of error from the mean:

\[ \text{Lowest Level} = 3324.3 - 2463.8 \]

\[ \text{Lowest Level} \approx 860.5 \]

Therefore, the lowest level of the 68% confidence interval estimate for the mean is approximately 860.5.