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
99% = mean ± 2.575 SEm
SEm = SD/√n
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.
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.