A business school placement director wants to estimate the...

A business school placement director wants to estimate the mean annual salaries 5 years after students graduate. A random sample of 25 such graduates found a sample mean of $$\$ 42,740$$ and a sample standard deviation of $$\$ 4,780$$. Find a $90 \%$ confidence interval for the population mean, assuming that the population distribution is normal.

Key Concepts

Margin of Error

The margin of error is the amount added to and subtracted from the point estimate to create the confidence interval. It quantifies the potential error in the estimate due to sampling variability and depends on the critical value from the t-distribution (or z-distribution) and the sample's standard deviation.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that can vary in an analysis once certain constraints (such as calculated sample statistics) are taken into account. In the context of constructing a confidence interval using the t-distribution, the degrees of freedom are typically equal to the sample size minus one.

Point Estimate

A point estimate is a single value derived from sample data used to estimate an unknown population parameter. In many cases, like estimating a mean, the sample mean serves as the point estimate, providing the center of the confidence interval.

Confidence Interval

A confidence interval is a range of values, based on sample data, that is likely to contain the true population parameter with a certain degree of confidence. It provides an estimate of the uncertainty associated with a sample statistic and gives researchers a set of plausible values for the parameter.

t-Distribution

The t-distribution is used in inferential statistics when the population standard deviation is unknown and the sample size is relatively small. It adjusts for the additional uncertainty in estimating the standard deviation and is characterized by degrees of freedom, which influence its shape.

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