Using Chebychev’s Theorem Old Faithful is a famous geyser at Yellowstone National Park. From a sample with n = 100, the mean interval between Old Faithful’s eruptions is 101.56 minutes and the standard deviation is 42.69 minutes. Using Chebychev’s Theorem, determine at least how many of the intervals lasted between 16.18 minutes and 186.94 minutes. (Adapted from Geyser Times)
Table of contents
- 1. Intro to Stats and Collecting Data55m
- 2. Describing Data with Tables and Graphs1h 55m
- 3. Describing Data Numerically1h 45m
- 4. Probability2h 16m
- 5. Binomial Distribution & Discrete Random Variables2h 33m
- 6. Normal Distribution and Continuous Random Variables1h 38m
- 7. Sampling Distributions & Confidence Intervals: Mean1h 53m
- 8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- 9. Hypothesis Testing for One Sample2h 19m
- 10. Hypothesis Testing for Two Samples3h 22m
- 11. Correlation1h 6m
- 12. Regression1h 4m
- 13. Chi-Square Tests & Goodness of Fit1h 20m
- 14. ANOVA1h 0m
3. Describing Data Numerically
Interpreting Standard Deviation
Problem 2.Q.6d
Textbook Question
Refer to the sample statistics from Exercise 5 and determine whether any of the house prices below are unusual. Explain your reasoning.
d. $147,000

1
Step 1: Recall the concept of unusual values in statistics. A value is considered unusual if it lies more than 2 standard deviations away from the mean. This is based on the empirical rule for normal distributions.
Step 2: Identify the mean (μ) and standard deviation (σ) of the house prices from the sample statistics provided in Exercise 5. These values are necessary to calculate the range of usual values.
Step 3: Calculate the lower and upper bounds for usual values using the formula: Lower Bound = μ - 2σ and Upper Bound = μ + 2σ. This will give the range within which most house prices are expected to fall.
Step 4: Compare the given house price of $147,000 to the calculated bounds. If $147,000 falls outside the range, it is considered unusual; otherwise, it is not unusual.
Step 5: Explain the reasoning based on the comparison. If $147,000 is outside the bounds, discuss how it deviates significantly from the mean, making it unusual. If it is within the bounds, explain that it is consistent with the expected variation in house prices.

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Unusual Values
In statistics, an unusual value, often referred to as an outlier, is a data point that significantly differs from the other observations in a dataset. Typically, values that lie beyond 1.5 times the interquartile range (IQR) above the third quartile or below the first quartile are considered unusual. Identifying unusual values helps in understanding the distribution and variability of the data.
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Step 3: Get P-Value
Descriptive Statistics
Descriptive statistics summarize and describe the main features of a dataset. Key measures include the mean, median, mode, range, variance, and standard deviation. These statistics provide insights into the central tendency and dispersion of the data, which are essential for determining whether specific values, like house prices, fall within a typical range.
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Parameters vs. Statistics
Normal Distribution
A normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In many cases, house prices can be assumed to follow a normal distribution, allowing for the application of statistical tests to identify unusual values based on standard deviations from the mean.
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Finding Standard Normal Probabilities using z-Table
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