Suppose we would like to estimate the mean amount of money spent on books by students in the fall semester. We have the following data from 400 randomly selected students: mean=$205 and s=$40. Assume that the amount spent on books by students is normally distributed. Compute a 99% confidence level for the population mean.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.005) and its Z score.

99% = mean ± 2.575 SEm

SEm = SD/√n

To compute the 99% confidence interval for the population mean, we can use the formula:

Confidence Interval = sample mean ± (critical value) × (standard deviation / √sample size)

First, let's determine the critical value for a 99% confidence level. Since our sample size is large (400), we can use a Z-table.

The critical value corresponding to a 99% confidence level is found by subtracting the desired confidence level from 1 and dividing that result by 2. In this case:

(1 - 0.99) / 2 = 0.005

Locating 0.005 in the Z-table, we find a corresponding Z-score of approximately 2.58.

Next, we substitute the given values into the formula to calculate the confidence interval:

Confidence Interval = $205 ± (2.58) × ($40 / √400)

Calculating the values:

Confidence Interval = $205 ± (2.58) × ($40 / 20)

Confidence Interval = $205 ± (2.58) × $2

Finally, we compute the lower and upper bounds of the confidence interval:

Lower bound = $205 - (2.58) × $2

Lower bound = $205 - $5.16

Lower bound ≈ $199.84

Upper bound = $205 + (2.58) × $2

Upper bound = $205 + $5.16

Upper bound ≈ $210.16

Therefore, the 99% confidence interval for the population mean amount of money spent on books by students in the fall semester is approximately $199.84 to $210.16.

To compute a 99% confidence level for the population mean, we can use the formula for confidence intervals:

Confidence Interval = mean ± (critical value) * (standard deviation / √n)

In this case, we have the following information:

Sample mean (x̄) = $205
Sample standard deviation (s) = $40
Sample size (n) = 400
Confidence level (1 - α) = 99%

First, we need to find the critical value corresponding to a 99% confidence level. The critical value is obtained from the z-table or a statistical software. For a 99% confidence level, the critical value is approximately 2.576.

Next, let's substitute the values into the formula:

Confidence Interval = $205 ± (2.576) * ($40 / √400)

To compute the margin of error, we divide the standard deviation by the square root of the sample size (√400) to account for the sample variation. In this case, √400 equals 20.

Confidence Interval = $205 ± (2.576) * ($40 / 20)

Simplifying further:

Confidence Interval = $205 ± (2.576) * $2

The margin of error is (2.576) * $2 = $5.152

Now, we can calculate the confidence interval:

Lower Limit = $205 - $5.152 = $199.848
Upper Limit = $205 + $5.152 = $210.152

Therefore, the 99% confidence interval for the population mean amount of money spent on books by students in the fall semester is approximately $199.85 to $210.15.