A random sample of greeting cards is a mean value of $2.55 and a standard deviation of $0.44 So construct a 95% confidence interval estimate of the mean value of all cards in the store's inventory. So if the 2500 cards in the inventory then how are the results in assisting the owner to estimate the total value of the inventory?

To construct a 95% confidence interval estimate of the mean value of all cards in the store's inventory, we can use the following formula:

Confidence Interval = Sample mean ± (Z * Standard Error)

Where:

- Sample mean is the mean value of the random sample (in this case, $2.55).
- Z is the critical value for the desired level of confidence. For a 95% confidence interval, the critical value (Z) is approximately 1.96.
- Standard Error is the standard deviation of the sample divided by the square root of the sample size. It represents the variability of the sample mean.

First, let's calculate the standard error:

Standard Error = Standard Deviation / √(Sample Size)
Standard Error = $0.44 / √(2500)

Now, plug in the values into the formula:

Confidence Interval = $2.55 ± (1.96 * Standard Error)

Calculate the standard error:

Standard Error = $0.44 / √(2500)
Standard Error ≈ $0.0088

Substitute it back into the formula:

Confidence Interval = $2.55 ± (1.96 * $0.0088)

Calculate the values:

Confidence Interval ≈ $2.55 ± $0.0172

Now, we can say with 95% confidence that the mean value of all the cards in the inventory falls within the range of approximately $2.5328 to $2.5672.

Regarding how these results assist the owner in estimating the total value of the inventory, the confidence interval provides a range within which the true mean value is likely to fall. In this case, the owner can be 95% confident that the mean value of all cards in the inventory is between approximately $2.5328 and $2.5672. To estimate the total value of the inventory, the owner can multiply the mean value by the number of cards in the inventory. In this case, the estimate would be:

Estimated total value of inventory = (Mean value of cards) * (Number of cards)
Estimated total value of inventory ≈ $2.55 * 2500
Estimated total value of inventory ≈ $6,375

So based on the confidence interval and estimated mean value, the owner can estimate that the total value of the inventory is approximately $6,375.