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Chapter 11: Problem 71

Decide whether the following statements are true or false. The mean and the median of the chi-square distribution are the same if \(d f=24\)

Short Answer

Expert verified
False, the mean and median are not the same for df=24.

Step by step solution

01

Understanding the Chi-Square Distribution

The chi-square distribution is a continuous probability distribution and it is used extensively in hypothesis testing and inferential statistics, such as in chi-square tests for variance and goodness-of-fit. It is characterized by the degrees of freedom (df), which also determines its shape.
02

Mean of the Chi-Square Distribution

The mean of the chi-square distribution is equal to the number of degrees of freedom. Therefore, if the degrees of freedom, denoted as df, is 24, the mean \(\mu\) is equal to 24.\[\mu = 24\]
03

Median of the Chi-Square Distribution

The median of the chi-square distribution is not exactly the same as the mean. While the mean is equal to the degrees of freedom, the median is given by an approximation formula: \( \text{median} \approx df \times \left(1 - \frac{2}{9df}\right)^3 \).For \(df = 24\), you calculate the median:\[\text{median} \approx 24 \times \left(1 - \frac{2}{9 imes 24}\right)^3 \approx 22.657.\]
04

Compare Mean and Median

The mean for a chi-square distribution with 24 degrees of freedom is 24, while the median calculated using the approximation is approximately 22.657. Therefore, since the mean is not equal to the median, the statement is false.

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Key Concepts

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

Degrees of Freedom
In statistics, the concept of degrees of freedom is crucial for defining the properties of many distributions, including the chi-square distribution.
Understanding degrees of freedom can help you grasp how these distributions behave as the parameter changes.
  • Degrees of freedom, often abbreviated as df, generally refer to the number of independent values or quantities that can vary in an analysis without breaking any constraints.
  • For the chi-square distribution, the degrees of freedom act as a parameter that defines the distribution's shape and spread.
  • The greater the degrees of freedom, the more the distribution approaches a normal distribution, showing increased data spread and peaking more at the mean.
For instance, if you have a chi-square distribution with 24 degrees of freedom, it will have specific characteristics like its mean being equal to 24. Understanding df allows you to fully utilize statistical tools like the chi-square tests for better data analysis.
Mean of Distribution
The mean of a probability distribution is the average expected value that you'd anticipate based purely on the distribution's characteristics.
In the chi-square distribution, the mean has a particularly direct relationship with the degrees of freedom.
  • The formula for the mean of a chi-square distribution is simple: it equals the degrees of freedom, \( \mu = df \).
  • This means that if you have a chi-square distribution with 24 degrees of freedom, the mean is precisely 24.
  • As a measure of central tendency, the mean gives insights into where most of the dataset's values will center around.
Knowing the mean can provide a quick glimpse into the expected outcome of your dataset, crucial for setting expectations in statistical hypothesis testing.
Median of Distribution
Unlike the mean, the median of a distribution might not be as straightforward to find, especially in non-symmetric distributions like the chi-square.
The median is the value separating the higher half of the data from the lower half, providing another form of central tendency.
  • The median for the chi-square distribution is often approximated because it doesn't always equal the mean, especially in non-normal shapes.
  • The approximation formula for the median is: \( \text{median} \approx df \times\left(1- \frac{2}{9df}\right)^3 \).This formula accounts for the skewness in the distribution.
  • For example, with 24 degrees of freedom, the median is roughly calculated as 22.657.
The difference between the mean and median highlights the chi-square distribution's skewness, affecting how you interpret hypothesis tests and fits against expected distributions.
Hypothesis Testing
The chi-square distribution is a cornerstone for hypothesis testing, particularly when dealing with categorical data.
Using this distribution allows statisticians to determine whether observed data diverges from expected data.
  • Hypothesis testing typically involves two hypotheses: the null hypothesis, which suggests no effect or relation, and the alternative hypothesis, which suggests the opposite.
  • With the chi-square distribution, hypothesis tests evaluate if an observed distribution matches a theoretical one, such as in goodness-of-fit tests.
  • Another application is testing the independence between variables in a contingency table, often through a chi-square test of independence.
Arming yourself with an understanding of the chi-square distribution in hypothesis testing gives you the ability to make informed, data-driven decisions by assessing statistical significance in your findings.

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Most popular questions from this chapter

An archer鈥檚 standard deviation for his hits is six (data is measured in distance from the center of the target). An observer claims the standard deviation is less. State the null and alternative hypotheses.

An article in the New England Journal of Medicine, discussed a study on smokers in California and Hawaii. In one part of the report, the self-reported ethnicity and smoking levels per day were given. Of the people smoking at most ten cigarettes per day, there were 9,886 African Americans, 2,745 Native Hawaiians, 12,831 Latinos, 8,378 Japanese Americans and 7,650 whites. Of the people smoking 11 to 20 cigarettes per day, there were 6,514 African Americans, 3,062 Native Hawaiians, 4,932 Latinos, 10,680 Japanese Americans, and 9,877 whites. Of the people smoking 21 to 30 cigarettes per day, there were 1,671 African Americans, 1,419 Native Hawaiians, 1,406 Latinos, 4,715 Japanese Americans, and 6,062 whites. Of the people smoking at least 31 cigarettes per day, there were 759 African Americans, 788 Native Hawaiians, 800 Latinos, 2,305 Japanese Americans, and 3,970 whites. Is this a right-tailed, left-tailed, or two-tailed test? Explain why.

Suppose an airline claims that its flights are consistently on time with an average delay of at most 15 minutes. It claims that the average delay is so consistent that the variance is no more than 150 minutes. Doubting the consistency part of the claim, a disgruntled traveler calculates the delays for his next 25 flights. The average delay for those 25 flights is 22 minutes with a standard deviation of 15 minutes. Is the traveler disputing the claim about the average or about the variance?

Which test do you use to decide whether an observed distribution is the same as an expected distribution?

a. Explain why a goodness-of-fit test and a test of independence are generally right-tailed tests. b. If you did a left-tailed test, what would you be testing?
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