A random sample of 70 observations from a normally distributed population possesses a sample mean equal to 26.2 and a sample standard deviation equal to 4.1.
a. Find an approximate 95% confidence interval for μ.
b. What do you mean when you say that a confidence coefficient is .95?
c. Find an approximate 99% confidence interval for μ.
d. What happens to the width of a confidence interval as the value of the confidence coefficient is increased while the sample size is held fixed?
e. Would your confidence intervals of parts a and c be valid if the distribution of the original population was not normal? Explain.
Step 1 of 5
Given a random sample of observations which follows a normally distributed population.
Given sample mean and sample standard deviation
We are asked to find an approximate confidence interval for
Let be a large sample of size from a population with mean and standard deviation
then a level confidence interval for is
Given sample mean , sample standard deviation and
Since we want 95% confidence interval, then a level is
Then for is , hence
Hence 95% confidence interval for is
Hence a 95% confidence interval for is .