A company is considering offering child care for their employees. They wish to
estimate the mean weekly child-care cost of their employees. A sample of 10 employees
reveals the following amounts spent last week in dollars.
107 92 97 95 105 101 91 99 95 104
Develop a 90% confidence interval for the population mean. Interpret the result.
=??, S=??, =??, Range of = (??) xt2/.
.
Find the mean first = sum of scores/number of scores
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation (SD) = square root of variance
Range = highest score - lowest
90% interval = mean ± 1.645SD
I'll let you do the calculations.
5s1a
company is considering offering child care for their employees. They wish to
estimate the mean weekly child-care cost of their employees. A sample of 10 employees
reveals the following amounts spent last week in dollars.
To develop a confidence interval for the population mean, we need to calculate the sample mean (x̄) and the sample standard deviation (s) first.
The sample mean (x̄) is the average of the sample data:
x̄ = (107 + 92 + 97 + 95 + 105 + 101 + 91 + 99 + 95 + 104) / 10 = 990 / 10 = 99
The sample standard deviation (s) measures the variability of the data points around the sample mean. To calculate the sample standard deviation, follow these steps:
1. Subtract the sample mean (x̄) from each data point.
(107 - 99) = 8
(92 - 99) = -7
(97 - 99) = -2
(95 - 99) = -4
(105 - 99) = 6
(101 - 99) = 2
(91 - 99) = -8
(99 - 99) = 0
(95 - 99) = -4
(104 - 99) = 5
2. Square each individual result.
(8)^2 = 64
(-7)^2 = 49
(-2)^2 = 4
(-4)^2 = 16
(6)^2 = 36
(2)^2 = 4
(-8)^2 = 64
(0)^2 = 0
(-4)^2 = 16
(5)^2 = 25
3. Sum up all the squared results.
64 + 49 + 4 + 16 + 36 + 4 + 64 + 0 + 16 + 25 = 278
4. Divide the sum by (n-1), where n is the sample size.
s = √(278 / (10-1)) ≈ √(278 / 9) ≈ √30.89 ≈ 5.55
Now that we have the sample mean (x̄) and sample standard deviation (s), we can proceed to calculate the confidence interval.
The formula to calculate the confidence interval is:
Confidence Interval = x̄ ± (t * (s / √n))
In this case, we want a 90% confidence interval, which means we need to determine the appropriate t-value for a sample size of 10 and a confidence level of 90%.
Using t-tables or statistical software, we find that the t-value for a 90% confidence level with 9 degrees of freedom is approximately 1.833.
Substituting the values into the formula:
Confidence Interval = 99 ± (1.833 * (5.55 / √10))
Calculating the margin of error:
Margin of Error = 1.833 * (5.55 / √10) ≈ 2.57
Substituting the margin of error into the confidence interval:
Confidence Interval = 99 ± 2.57
Interpretation:
The 90% confidence interval for the mean weekly child-care cost of the employees is (96.43, 101.57). This means that, based on the sample data, we are 90% confident that the true population mean falls within this range.