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Yellowstone National Park: Old Faithful Geyser 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) Check Requirements 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) Interpretation 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

Review sample sizes and distribution selection

For statistical inference regarding the difference in means, the normal distribution can be used provided the sample sizes are large. In this scenario, the sample sizes are 9340 and 25111, respectively, which are considered large. Therefore, we use a normal distribution to approximate the distribution of \( \bar{x}_{1} - \bar{x}_{2} \).

02

Set up confidence interval formula

The formula for a confidence interval for the difference in means is:\[(\bar{x}_{1} - \bar{x}_{2}) \pm z_{\alpha/2} \times \sqrt{\frac{\sigma_{1}^2}{n_{1}} + \frac{\sigma_{2}^2}{n_{2}}}\]where \( z_{\alpha/2} \) is the critical value from the normal distribution table for a 99% confidence level.

03

Find the critical value

For a 99% confidence level, \( \alpha = 0.01 \) so \( \alpha/2 = 0.005 \). Using a normal distribution table, find that \( z_{0.005} \approx 2.576 \).

04

Calculate the standard error

Compute the standard error using the given standard deviations and sample sizes:\[\sqrt{\frac{9.17^2}{9340} + \frac{12.67^2}{25111}} \approx 0.1538\]

05

Calculate the confidence interval

Substitute the values into the confidence interval formula:\[(63.3 - 72.1) \pm 2.576 \times 0.1538\]\[\\Rightarrow -8.8 \pm 2.576 \times 0.1538\]\[\\Rightarrow -8.8 \pm 0.3966\]This gives the interval: \(-9.1966, -8.4034\).

06

Interpret the confidence interval

The confidence interval \(-9.1966, -8.4034\) consists only of negative values, implying that the mean interval time between eruptions was longer in the 1983-1987 period compared to the 1948-1952 period at the 99% confidence level. This suggests that a change in the interval length between eruptions likely occurred.

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Key Concepts

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

Normal Distribution

The normal distribution, often referred to as the bell curve due to its distinctive shape, is a foundational concept in statistics.
It describes a lot of naturally occurring data, where values are symmetrically distributed around a mean.
The mean is the peak of the curve and measures the average of a data set, while the standard deviation determines the spread of values.

• If the values are concentrated around the mean, the standard deviation is low, and the curve is narrow.
• If they spread out widely, the standard deviation is high, making the curve wider.

In our exercise, utilizing the normal distribution was possible due to the large sample sizes from Old Faithful's data in different years.
According to statistical principles, if a sample size is sufficiently large (typically n>30), the sample means will approximately follow a normal distribution even if the original data is not perfectly normal.
This property is known as the Central Limit Theorem, and it often simplifies the analysis of complex data sets when the sample size is large.

Sample Mean

In statistical terms, the sample mean, denoted by \(\bar{x}\), is the average value of a set of data points collected from a larger population.
It's an estimate of the population mean, which represents the true average.
In our Old Faithful Geyser context, the sample means \(\bar{x}_{1}=63.3\) and \(\bar{x}_{2}=72.1\) are used to analyze the behavior of the geyser over different periods.
Calculating the sample mean involves summing all the individual data points and dividing the total by the number of points.

• The formula is \(\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_{i}\), where \(n\) is the number of observations.

By comparing sample means from two different time periods, one can infer changes or trends in the underlying phenomena, such as the interval times between geyser eruptions.
Understanding sample means thus provides insights into larger population characteristics without examining every instance.

Population Mean

The population mean, represented by \( \mu \), is a critical measure in statistics that reflects the average of an entire data set.
It is considered a parameter of the population and represents the true center of the data distribution.
Unlike the sample mean, which is derived from a subset of data, the population mean takes into account every possible data point within a population.
Calculating it involves the formula \( \mu = \frac{1}{N} \sum_{i=1}^{N} x_{i} \), where \( N \) is the total number of data points in the population.
In the context of the exercise, \( \mu_{1} \) and \( \mu_{2} \) indicate the true average time interval between eruptions for each period analyzed.
Knowing the population mean offers the most accurate understanding of data trends and patterns, but obtaining it can be challenging if the entirety of the data set is not available.
Therefore, inferential statistics often rely on sample means and confidence intervals to predict and understand the population mean.

Standard Error

The standard error is a statistical concept that measures the accuracy with which a sample represents a population.
It indicates how much variability exists in a sample's mean as an estimate of the population mean.
A smaller standard error points to a more precise estimate, while a larger standard error suggests more variability and less precision.
In the confidence interval calculation for Old Faithful's eruption times, the standard error was calculated as \( \sqrt{\frac{9.17^2}{9340} + \frac{12.67^2}{25111}} \approx 0.1538 \).
This value reflects how much the sample means \( \bar{x}_{1} \) and \( \bar{x}_{2} \) could vary from the true population means \( \mu_{1} \) and \( \mu_{2} \).
Essentially, standard error helps in defining the confidence interval of the mean estimate, allowing us to express the range where the true population mean is likely to fall.
It reinforces the reliability of the inferences made about population parameters, highlighting any potential discrepancies between sampled data and the overall population.