In a survey randomly selected subjects, the mean age of the 36 respondents is 40.0 years and the standard deviation of the ages is 10.0 years. Use these sample results to construct a 95% confidence interval estimate of the mean age of the population from which the sample was selected.
95% confidence means that you will have 5% left in the two tails. Split this number in half to get .025 in each tail.
This is equivalent to a z-score of 1.96
Use this z-score in the formula to find the confidence interval.
To construct a 95% confidence interval estimate of the mean age of the population, we can use the following formula:
Confidence Interval = Sample Mean ± (Critical Value) * (Standard Deviation /√Sample Size)
The critical value can be found using a t-distribution table or a calculator based on the given level of confidence and degrees of freedom. Since the sample size is 36, the degrees of freedom will be 35.
The critical value for a 95% confidence interval with 35 degrees of freedom is approximately 2.030.
Plugging in the given values into the formula, we get:
Confidence Interval = 40.0 ± (2.030) * (10.0 / √36)
Now we can calculate the confidence interval:
Confidence Interval = 40.0 ± (2.030) * (10.0 / 6)
Simplifying the expression, we get:
Confidence Interval = 40.0 ± (2.030) * (1.667)
Confidence Interval = 40.0 ± 3.386
Therefore, the 95% confidence interval estimate of the mean age of the population is (36.614, 43.386).
To construct a 95% confidence interval estimate of the mean age of the population from which the sample was selected, we can use the following formula:
Confidence interval = sample mean ± (critical value * standard error)
First, let's calculate the critical value. Since the sample size is relatively small (n = 36), we need to use a t-distribution table. For a 95% confidence level and 35 degrees of freedom (n-1), the critical value can be found at t-value = 2.03.
Next, let's calculate the standard error using the formula:
Standard error = standard deviation / √(sample size)
For this sample, the standard deviation is 10.0 years and the sample size is 36.
Standard error = 10.0 / √(36) = 10.0 / 6 = 1.67
Now, we can plug the values into the confidence interval formula:
Confidence interval = 40.0 ± (2.03 * 1.67)
Confidence interval = 40.0 ± 3.39
Finally, we can calculate the lower and upper confidence limits:
Lower limit = 40.0 - 3.39 = 36.61
Upper limit = 40.0 + 3.39 = 43.39
Therefore, the 95% confidence interval estimate of the mean age of the population from which the sample was selected is between 36.61 years and 43.39 years.