A senior manager, responsible for a group of 120 junior...

A senior manager, responsible for a group of 120 junior executives, is interested in the total amount of time per week spent by these people in internal meetings. A random sample of 35 of these executives was asked to keep diary records during the next week. When the results were analyzed, it was found that these sample members spent a total of 143 hours in internal meetings. The sample standard deviation was 3.1 hours. Find a $90 \%$ confidence interval for the total number of hours spent in internal meetings by all 120 junior executives in the week.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the value of an unknown population parameter. It quantifies the uncertainty of an estimate and is characterized by a confidence level, which represents the proportion of all possible samples that can be expected to include the true parameter value.

t-distribution

The t-distribution is a probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. It is used when the sample size is relatively small and the population standard deviation is unknown. The t-distribution accounts for additional uncertainty in the estimate of the population standard deviation.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation of a statistic that are free to vary. In the context of estimating a parameter from a sample, the degrees of freedom affect the shape of the t-distribution. Typically, it is equal to the sample size minus one when calculating the sample standard deviation.

Scaling Factor for Total Estimation

When estimating a population total from a sample, the sample mean is scaled up by the ratio of the population size to the sample size. This extrapolation is used to estimate the total quantity for the entire population based on the average observed in the sample.

Sample Mean and Standard Deviation

The sample mean is the arithmetic average of the data collected from the sample, and it serves as an unbiased estimate of the population mean. The sample standard deviation measures the dispersion of the data around the sample mean, providing an estimate of the population variability when the true population standard deviation is unknown.

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