solve: our health organization is interested in estimating the average value of outstanding accounts receivable. After selecting a sample of 100 accounts, suppose that the mean was $120 and the standard deviation was $40. Use these data to construct a 98% confidence interval. how do i solve?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.01 for two-tailed) and its Z score (score in terms of standard deviations).
98% = mean ± 2.33 SEm
SEm = SD/√n
To construct a confidence interval, you can follow these steps:
Step 1: Identify the sample mean and standard deviation given in the problem.
Sample mean (x̄) = $120
Standard deviation (σ) = $40
Sample size (n) = 100
Step 2: Determine the confidence level.
In this case, the confidence level is 98%. This means that we want to construct a confidence interval where we are 98% confident that the interval captures the true population mean.
Step 3: Find the critical value corresponding to the confidence level.
The critical value depends on the sample size and the desired confidence level. Since the sample size is large (n > 30), we can use the standard normal distribution (z-distribution). For a 98% confidence level, we need to find the z-value that leaves 1% in each tail of the distribution.
Using a standard normal distribution table or calculator, the critical z-value for a 98% confidence level is approximately 2.33.
Step 4: Calculate the margin of error.
The margin of error is a measure of the uncertainty in our estimate. It indicates the range within which the true population mean is likely to fall. The formula for calculating the margin of error is:
Margin of Error (E) = (critical value) * (standard deviation) / sqrt(sample size)
E = 2.33 * 40 / sqrt(100) ≈ 9.32
Step 5: Construct the confidence interval.
Finally, we can construct the confidence interval by adding and subtracting the margin of error from the sample mean:
Confidence Interval = (sample mean) ± (margin of error)
CI = $120 ± $9.32
CI ≈ ($110.68, $129.32)
So, the 98% confidence interval for the average value of outstanding accounts receivable is approximately $110.68 to $129.32.