A studyis run to estimate the mean total cholesterol level in children 2 to 6 years of age. A sample of 9 participants is selected and their total cholesterol levels are measured as follows:
185 225 240 196 175 180 194 147 223
Generate a 95% confidence interval for the true mean total cholesterol leels in children.
95% confidence interval = mean ± 1.96 SEm
SEm = SD/√n
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 = square root of variance
I'll let you do the calculations.
To generate a confidence interval for the true mean total cholesterol levels in children, we can use the formula:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
1. Calculate the sample mean:
Add up all the cholesterol levels and divide the sum by the number of participants (in this case, 9).
Sample Mean = (185 + 225 + 240 + 196 + 175 + 180 + 194 + 147 + 223) / 9
2. Calculate the standard deviation of the sample:
Subtract the sample mean from each data point, square the differences, add up all the squared differences, divide by the number of participants minus 1, and take the square root.
Standard Deviation = √[( (185 - Sample Mean)² + (225 - Sample Mean)² + ... + (223 - Sample Mean)² ) / (9 - 1)]
3. Calculate the standard error:
Divide the standard deviation by the square root of the number of participants.
Standard Error = Standard Deviation / √(Number of Participants)
4. Determine the critical value:
For a 95% confidence interval, we need to find the critical value from the t-table or calculator. The degrees of freedom for this problem is 9 - 1 = 8.
Using a t-table or calculator, the critical value for a 95% confidence interval with 8 degrees of freedom is approximately 2.306.
5. Calculate the confidence interval:
Multiply the critical value by the standard error, and then add/subtract the result from the sample mean.
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Now, let's do the calculation:
Step 1: Calculate the Sample Mean:
Sample Mean = (185 + 225 + 240 + 196 + 175 + 180 + 194 + 147 + 223) / 9 = 198.11
Step 2: Calculate the Standard Deviation:
Standard Deviation = √[( (185 - 198.11)² + (225 - 198.11)² + ... + (223 - 198.11)² ) / (9 - 1)]
Standard Deviation = 29.129
Step 3: Calculate the Standard Error:
Standard Error = Standard Deviation / √(9) = 29.129 / √(9) = 9.71
Step 4: Determine the Critical Value:
Using a t-table or calculator, the critical value for a 95% confidence interval with 8 degrees of freedom is approximately 2.306.
Step 5: Calculate the Confidence Interval:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
Confidence Interval = 198.11 ± (2.306 * 9.71)
Confidence Interval ≈ (177.02, 219.20)
Therefore, the 95% confidence interval for the true mean total cholesterol levels in children 2 to 6 years of age is approximately (177.02, 219.20).