Final answer:
The 95% confidence interval for the mean height of Mike's friends is calculated to be between 64.197 inches and 72.663 inches.
Step-by-step explanation:
To calculate the 95% confidence interval for the mean height of Mike's friends, we need to know the sample mean, the sample standard deviation, and the sample size. Since we have the sample standard deviation (4.577) and the sample size (n = 7), we can calculate the sample mean as follows:
- Sum the heights: 65+71+74+61+66+70+72 = 479 inches.
- Divide by the number of friends: 479 / 7 = 68.43 inches (The sample mean).
To construct a 95% confidence interval for the mean height, we will use the t-distribution since the sample size is small and we're assuming we do not know the population standard deviation. We need to find the t-value that corresponds to a 95% confidence level for a sample size of 7, which has 6 degrees of freedom (n-1).
Using statistical tables or software, we find that the t-value is approximately 2.447 for a 95% confidence level with 6 degrees of freedom. The formula for the confidence interval is given by:
Confidence interval = sample mean ± (t-value × (sample standard deviation / √n))
Plugging the values we have:
Confidence interval = 68.43 ± (2.447 × (4.577 / √7))
This calculation results in the interval:
68.43 ± (2.447 × 1.730) = 68.43 ± 4.233 = (64.197, 72.663)
Therefore, we are 95% confident that the true mean height of Mike's friends lies between 64.197 inches and 72.663 inches.