Final answer:
To construct a confidence interval for a mean of 20 with a sample size of 64 and a sample standard deviation of 4, we use the t-distribution with 60 degrees of freedom. Assuming a 95% confidence level with a t-value of approximately 2.00, the confidence interval is (19, 21), indicating with confidence that the population mean is between these two values.
Step-by-step explanation:
To construct a confidence interval for the mean when the population standard deviation is unknown and the sample size is 64, we use the sample standard deviation of 4 and the degrees of freedom for 60. This is done by utilizing the t-distribution since the sample standard deviation is used as an estimator for the population standard deviation.
First, identify the confidence level; if it is not specified, typically a 95% confidence level is used. Then, find the corresponding t-value from the t-distribution table. Next, insert the sample mean, the t-value, the sample standard deviation, and the sample size into the formula to calculate the confidence interval.
The degrees of freedom in this case would be the sample size minus one, which is 63. However, for practical purposes, when using statistical tables, we use the closest degree of freedom available, which is 60 in this case.
Suppose we are constructing a 95% confidence interval and the t-value for 60 degrees of freedom is approximately 2.00.
The confidence interval is (19, 21). This means we estimate with 95% confidence that the true population mean is between 19 and 21.