91Ó°ÊÓ

The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to \(1987 .\) Some of these data are published in the book The Story of Old Faithful, by G. D. Marler (Yellowstone Association Press). Let \(x_{1}\) be a random variable that represents the time interval (in minutes) between Old Faithful's eruptions for the years 1948 to \(1952 .\) Based on 9340 observations, the sample mean interval was \(\bar{x}_{1}=63.3\) minutes. Let \(x_{2}\) be a random variable that represents the time interval in minutes between Old Faithful's eruptions for the years 1983 to \(1987 .\) Based on 25,111 observations, the sample mean time interval was \(\bar{x}_{2}=72.1\) minutes. Historical data suggest that \(\sigma_{1}=9.17\) minutes and \(\sigma_{2}=12.67\) minutes. Let \(\mu_{1}\) be the population mean of \(x_{1}\) and let \(\mu_{2}\) be the population mean of \(x_{2}\) (a) Which distribution, normal or Student's \(t,\) do we use to approximate the \(\bar{x}_{1}-\bar{x}_{2}\) distribution? Explain. (b) Compute a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) (c) Comment on the meaning of the confidence interval in the context of this problem. Does the interval consist of positive numbers only? negative numbers only? a mix of positive and negative numbers? Does it appear (at the \(99 \%\) confidence level) that a change in the interval length between eruptions has occurred? Many geologic experts believe that the distribution of eruption times of Old Faithful changed after the major earthquake that occurred in 1959

Step by step solution

01

Determine the Appropriate Distribution

To decide whether to use the normal distribution or Student's t-distribution, we must consider the sample size and population standard deviation. Since both sample sizes 9340 and 25111 are larger than 30, and we know the population standard deviations \( \sigma_{1} = 9.17 \) and \( \sigma_{2} = 12.67 \), we use the normal distribution according to Central Limit Theorem.

02

Calculate Standard Error of Difference in Means

The standard error of the difference in means is given by the formula: \[ SE_{\bar{x}_{1} - \bar{x}_{2}} = \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}} \]Substitute the values: \[ SE_{\bar{x}_{1} - \bar{x}_{2}} = \sqrt{\frac{9.17^{2}}{9340} + \frac{12.67^{2}}{25111}} \] Calculate \( SE_{\bar{x}_{1} - \bar{x}_{2}} \).

03

Compute the Z-value for 99% Confidence Interval

For a 99% confidence interval, the Z-value is approximately 2.576 (this corresponds to the area in both tails being 0.01/2 = 0.005).

04

Calculate the Margin of Error

The margin of error \( E \) is calculated as: \[ E = Z \times SE_{\bar{x}_{1} - \bar{x}_{2}} \] Substitute the Z-value and standard error calculated previously to find the margin of error.

05

Construct the Confidence Interval

The 99% confidence interval for \( \mu_{1} - \mu_{2} \) is given by: \[ (\bar{x}_{1} - \bar{x}_{2}) \pm E \] Substitute \( \bar{x}_{1} = 63.3 \), \( \bar{x}_{2} = 72.1 \), and the calculated margin of error \( E \) from Step 4.

06

Interpret the Confidence Interval

Examine the confidence interval to determine if it consists exclusively of positive, exclusively of negative numbers, or a mix. The interval's sign indicates whether there has likely been a change in the mean eruption interval times. A negative interval or one that includes zero suggests no significant change at the 99% confidence level.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normal Distribution

In statistics, the normal distribution is a highly important concept, often referred to as the bell curve due to its shape. It is symmetrical and most of the data points lie close to the mean. The tails taper off equally on both sides. When we have large sample sizes, as seen in the Old Faithful eruption data case, the sample means tend to approximate a normal distribution even if the data itself does not. This happens due to the magic of the Central Limit Theorem.

Using the normal distribution in the context of the Old Faithful exercise, we are able to model the difference in average eruption intervals between two different time periods. Since the sample sizes (9340 and 25111) are sufficiently large, the distribution of the sample mean should follow a normal distribution, allowing us to use it to calculate confidence intervals.

Central Limit Theorem

The Central Limit Theorem (CLT) is a cornerstone of statistics. It states that if you take enough samples of a population mean, the distribution of those sample means will be normal (or nearly normal), regardless of the shape of the original data distribution.

In practical terms, for the Old Faithful data, we rely on the CLT to assume normality in the difference of means between the two times the geyser was observed. This is extremely useful because it lets us apply inferential statistical methods, like calculating confidence intervals, even when the data may not inherently be normal, as long as the sample size is large enough.

Standard Error of the Difference in Means

The standard error of the difference in means quantifies how much variability one can expect in the difference between two sample means. It relies on both the standard deviations of the samples and the sizes of these samples.

For the Old Faithful dataset, the calculation of this standard error helps us determine how much uncertainty there is in comparing eruption intervals between the two periods studied. The formula for calculating it is:

• \[ SE_{\bar{x}_{1} - \bar{x}_{2}} = \sqrt{\frac{\sigma_{1}^{2}}{n_{1}} + \frac{\sigma_{2}^{2}}{n_{2}}} \]

This equation takes into account the variability (standard deviation \( \sigma \)) and the sample sizes \( n \) of each observational period, allowing us to compute the standard error in the mean differences.

Margin of Error

The margin of error allows us to establish a range around the sample mean that we can be fairly certain comprises the true population mean. It reflects the degree of imprecision in our sample estimate.

For Old Faithful's eruption data, the margin of error is derived from multiplying the z-score for a desired confidence level, such as 99%, with the standard error. The calculation provides insight into how much the sample mean might vary from the true mean.

To calculate the margin of error, the following formula is used:

• \[ E = Z \times SE_{\bar{x}_{1} - \bar{x}_{2}} \]

Where \( Z \) is the z-value corresponding to the confidence level of interest (2.576 for 99%).
This margin can significantly affect how we interpret changes in eruption intervals over time, offering evidence of shifts or stability in eruption patterns.