Q. 7.51 - Elementary Statistics

Question

According to The Earth: Structure, Composition and Evolution for earthquakes with a magnitude of 7.5 or greater on the Richter scale, the time between successive earthquakes has a mean of 437 days and a standard deviation of 399 days. Suppose that you observe a sample of four times between successive earthquakes that have a magnitude of 7.5 or greater on Richter scale.

Part (a): On average, what would you expect to be the mean of the four times?

Part (b): How much variation would you expect from your answer in part (a)?

Step-by-Step Solution

Answer

Part (a): On an average the mean of the four times in the sample will be equal to 437 days since the mean of all possible sample mean is equal to population mean.

Part (b): The variation that is expected from the answer in part (a) is $\pm 598.5$days.

Part (a) Step 1. Given information.

Consider the given question,

Population consists of earthquakes with magnitude of 7.5 hours a greater on Richter scale.

Population mean time between two earthquakes $\mu =437$days.

Population of standard deviation,$\sigma =399$ days.

The sample size is $n=4$.

Part (a) Step 2. Determine the mean of the four times.

Assume the sample mean time between two earthquakes is denoted by $\overline{x}$.

Then, mean of $$\begin{gathered} \overline{x}=\mu \\ =437days \end{gathered}$$

This mean, on an average we can expect that mean of the four times in the sample will be equal to 437 days since the mean of all possible sample mean is equal to population mean $\mu$.

Part (b) Step 1. Determine the variation.

Standard deviation of all possible mean ${\sigma }_{\overline{x}}$,

$$\begin{gathered} =\frac{\sigma }{\sqrt{n}} \\ =\frac{399}{\sqrt{4}} \\ =199.5 days \end{gathered}$$

We can expect $\pm 3{\sigma }_{\overline{x}}$ variation from the mean of $\overline{x}$, i.e., from population mean $\mu$.

That is expected amount of variation from,

$$\begin{gathered} \overline{x}=\pm 3{\sigma }_{\overline{x}} \\ =\pm 3\frac{\sigma }{\sqrt{n}} \\ =\pm 3\times 199.5 \\ =\pm 598.5 \end{gathered}$$