Question

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.

Short answer

Question: Estimate the mean angular direction of the fractures and find the margin of error for this estimate, given a sample of 50 fractures, with a sample mean of 39.8° and a sample standard deviation of 17.2°, using a 95% confidence level. Answer: The estimated mean angular direction of the fractures lies within the interval (34.91°, 44.69°) with a margin of error of approximately ±4.89° and a 95% confidence level.

Step-by-step solution

Determine the degrees of freedom and find the t-value

To determine the degrees of freedom for the t-distribution, subtract 1 from the sample size, i.e., \(df = n-1 = 50 - 1 = 49\). Now, we need to find the t-value for a 95% confidence level. You can check a t-distribution table or use a calculator or software that gives you the appropriate t-value. For a two-tailed test with \(df=49\) and a confidence level of 95%, we find \(t = 2.0096\).

Calculate the Standard Error

The standard error (SE) is the standard deviation of the sampling distribution and can be calculated using the formula: $$ SE = \frac{s}{\sqrt{n}} $$ Where s is the sample standard deviation \((17.2^{\circ})\) and n is the sample size \((50)\). Plugging in the values, we get: $$ SE = \frac{17.2}{\sqrt{50}} \approx 2.436 $$

Calculate the Margin of Error

The margin of error (ME) is the range within which the true population mean lies. It can be calculated using the formula: $$ ME = t \times SE $$ Where t is the t-value for the desired confidence level and SE is the standard error calculated in Step 2. Plugging in the values, we get: $$ ME = 2.0096 \times 2.436 \approx 4.89 $$

Calculate the Confidence Interval

Now we can calculate the 95% confidence interval for the mean angular direction of the fractures using the sample mean and the margin of error. The confidence interval is given by the range: $$ (\bar{x} - ME, \bar{x} + ME) $$ Where \(\bar{x}\) is the sample mean. Plugging in the values, we get: $$ (39.8 - 4.89, 39.8 + 4.89) \approx (34.91^{\circ}, 44.69^{\circ}) $$ The estimated mean angular direction of the fractures lies within the interval \((34.91^{\circ}, 44.69^{\circ})\) with a margin of error of approximately \(\pm4.89^{\circ}\) and a 95% confidence level.

Key concepts

Degrees of Freedom

In statistical analysis, particularly in the context of estimating population parameters, the term 'degrees of freedom' (df) is crucial. It represents the number of independent values in a statistical calculation that are free to vary. To calculate it, one usually subtracts the number of parameters estimated from the number of data points. For instance, if you have a sample of 50 measurements and you are estimating one parameter, such as the mean, the degrees of freedom would be 49 (
\(df = n - 1 = 50 - 1 = 49\)). The concept is vital when we use sampling distributions like the t-distribution to estimate population parameters because the shape of the t-distribution and, in turn, critical values we use for confidence intervals or hypothesis testing depend heavily on the degrees of freedom.

Standard Error

The standard error is a powerful statistical tool that measures the precision of an estimate. Specifically, it describes the standard deviation of the sampling distribution of a statistic. To compute the standard error of the mean, for instance, you'd take the sample standard deviation (s), which for our geological study was 17.2 degrees, and divide it by the square root of the sample size (n), which was 50 fractures. The formula is therefore:
\[SE = \frac{s}{\sqrt{n}}\], resulting in \(SE\approx 2.436\). The standard error helps quantify the variability one might expect in the sample mean if a different sample were taken, reflecting the accuracy of the sample mean as an estimator for the population mean.

T-Distribution

The t-distribution is pivotal in small sample statistics, often used when the population standard deviation is unknown and the sample size is small. Unlike the normal distribution, which is more bell-shaped and has thinner tails, the t-distribution has thicker tails, indicating a higher probability for extreme values. This makes it more appropriate for small samples where extreme values can have a larger impact. As the sample size grows, the t-distribution approximates the normal distribution. In our exercise, we used a t-distribution because we were working with a sample mean and sample size of 50, which required us to consider the variability that comes with less information than the population would provide. The distribution's shape depends on the degrees of freedom; as degrees of freedom increase, it more closely resembles the normal distribution.

Margin of Error

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It represents the radius of an interval and is used to create the confidence intervals for estimates. When estimating the mean angle of fractures, the margin of error signifies how far we expect the true population mean to be from our sample mean. To calculate it, as shown in the geological study, you multiply the t-value, which correlates with the desired confidence level and degrees of freedom (in our case, 2.0096), by the standard error:
\[ME = t \times SE\]. This operation yielded a margin of error of approximately 4.89 degrees. Therefore, we are 95% confident that the population mean falls within an interval defined by this margin of error.

Sample Mean

The sample mean is the average value in a sample and serves as an estimator of the population mean. It is calculated by summing all the measurements in the sample and dividing by the number of observations. Represented as \(\bar{x}\), it can vary from sample to sample due to the randomness inherent in sampling, which this variability is what makes calculating measures of uncertainty, like the standard error and the margin of error, so important. In the context of our geological example, the sample mean was 39.8 degrees. By establishing the confidence interval around this sample mean, we have identified a range within which we are quite certain the true population mean lies.