A coffee house weighs 10 cups of coffee for customers and finds a mean of 12.5 ounces and a standard deviation of 0.5 ounces. Assuming these cups are a random sample of all cups, which method would be used to estimate the mean weight of all cups?

A coffee house weighs 10 cups of coffee for customers and finds a mean of 12.5 ounces and a standard deviation of 0.5 ounces. Assuming these cups are a random sample of all cups, which method would be used to estimate the mean weight of all cups?

• A. 2-sample t-interval
• B. Z-interval
• C. 1-sample t-interval with 9 degrees of freedom
• D. Paired t-interval

Answer: (C)

Estimating a population mean from a small random sample when the population standard deviation is unknown uses a one-sample t-interval. With ten cups, the sample size is small, so we use the t distribution with degrees of freedom equal to n−1, which is nine in this case. The interval centers on the sample mean (12.5 ounces) and uses the sample standard deviation (0.5 ounces) to determine the margin of error, reflecting the extra variability you’d expect when sigma isn’t known and the sample is small. A z-interval would require knowing the population standard deviation, which isn’t provided here. A two-sample or paired t-interval is for comparing two groups or paired data, not for estimating the mean from a single sample. So the appropriate method is a one-sample t-interval with 9 degrees of freedom.

Estimating a population mean from a small random sample when the population standard deviation is unknown uses a one-sample t-interval. With ten cups, the sample size is small, so we use the t distribution with degrees of freedom equal to n−1, which is nine in this case. The interval centers on the sample mean (12.5 ounces) and uses the sample standard deviation (0.5 ounces) to determine the margin of error, reflecting the extra variability you’d expect when sigma isn’t known and the sample is small. A z-interval would require knowing the population standard deviation, which isn’t provided here. A two-sample or paired t-interval is for comparing two groups or paired data, not for estimating the mean from a single sample. So the appropriate method is a one-sample t-interval with 9 degrees of freedom.