A geologist studying the movement of the earth's crust at a...

A geologist studying the movement of the earth's crust at a particular location on California's San Andreas fault found many fractures in the local rock structure. In an attempt to determine the mean angle of the breaks, she sampled $n=50$ fractures and found the sample mean and standard deviation to be $39.8^{\circ}$ and $17.2^{\circ},$ respectively. Estimate the mean angular direction of the fractures and find the margin of error for your estimate.

Key Concepts

Point Estimation

Point estimation involves using sample data to calculate a single value that serves as the best guess of an unknown population parameter. In this context, the sample mean is used as a point estimate of the true (population) mean angular direction of the fractures.

Standard Error of the Mean

The standard error of the mean quantifies the dispersion of the sample mean estimates around the true population mean. It is calculated by dividing the sample standard deviation by the square root of the sample size, reflecting how precise the sample mean is as an estimate of the population mean.

Confidence Interval

A confidence interval provides a range of plausible values for the population parameter based on the sample data and a specified level of confidence (often 95%). It gives both an estimate and an indication of the uncertainty associated with that estimate.

Margin of Error

The margin of error represents half the width of a confidence interval and quantifies the extent of the uncertainty surrounding the point estimate. It is obtained by multiplying the appropriate critical value from the t-distribution (or normal distribution when applicable) by the standard error.

Student's t-Distribution

When the population standard deviation is unknown and the sample size is relatively small, the Student's t-distribution is used to determine the critical value for constructing the confidence interval. The t-distribution accounts for additional uncertainty due to estimation of the standard deviation, with the precise shape of the distribution determined by the degrees of freedom (typically n-1).

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