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Chapter 12: Problem 1

Describe the main characteristics of an \(F\) distribution.

Short Answer

Expert verified
An F distribution is a probability distribution commonly used in statistics for hypothesis testing, especially for comparing variances of two populations. It is positively skewed, and its shape depends on the degrees of freedom. A crucial thing to remember about F distribution is that it has two sets of degrees of freedom - one for the numerator and one for the denominator of the F ratio.

Step by step solution

01

F Distribution Definition

The F distribution is a type of statistical distribution that is related to the chi-square distribution and arises in the ANOVA, amongst others. It is often used in hypothesis testing when comparing variances of two populations.
02

Shape of the F Distribution

The F distribution is described in terms of degrees of freedom. It can only take positive values and is not symmetric, instead it is positively skewed. It starts at zero, increases, and then decreases. The exact shape depends on the degrees of freedom.
03

Degrees of Freedom

The concept of degrees of freedom is also integral to the F distribution. It has two sets of degrees of freedom鈥攐ne for each of the variances being compared. The first set pertains to the degrees of freedom in the numerator of the F ratio, and the second set pertains to the degrees of freedom in the denominator of the F ratio.
04

Usage

In general, the F-distribution is used to test whether two independent estimates of variance can be assumed to be estimates of the same variance.

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Most popular questions from this chapter

Find the critical value of \(F\) for an \(F\) distribution with \(.025\) area in the right tail and a. \(d f=(4,11)\) b. \(d f=(15,3)\)

The following table lists the numbers of violent crimes reported to police on randomly selected days for this year. The data are taken from three large cities of about the same size. $$ \begin{array}{rrr} \hline \text { City A } & \text { City B } & \text { City C } \\ \hline 5 & 2 & 8 \\ 9 & 4 & 12 \\ 12 & 1 & 10 \\ 3 & 13 & 3 \\ 9 & 7 & 9 \\ 7 & 6 & 14 \\ 13 & & \\ \hline \end{array} $$ Using a \(5 \%\) significance level, test the null hypothesis that the mean number of violent crimes reported per day is the same for each of these three cities.

A university employment office wants to compare the time taken by graduates with three different majors to find their first fulltime job after graduation. The following table lists the time (in days) taken to find their first full- time job after graduation for a random sample of eight business majors, seven computer science majors, and six engineering majors who graduated in May 2014 . $$ \begin{array}{ccc} \hline \text { Business } & \text { Computer Science } & \text { Engineering } \\\ \hline 208 & 156 & 126 \\ 162 & 113 & 275 \\ 240 & 281 & 363 \\ 180 & 128 & 146 \\ 148 & 305 & 298 \\ 312 & 147 & 392 \\ 176 & 232 & \\ 292 & & \\ \hline \end{array} $$ At a \(5 \%\) significance level, can you reject the null hypothesis that the mean time taken to find their first full-time job for all May 2014 graduates in these fields is the same?

An ophthalmologist is interested in determining whether a golfer's type of vision (far-sightedness, near-sightedness, no prescription) impacts how well he or she can judge distance. Random samples of golfers from these three groups (far-sightedness, near-sightedness, no prescription) were selected, and these golfers were blindfolded and taken to the same location on a golf course. Then each of them was asked to estimate the distance from this location to the pin at the end of the hole. The data (in yards) given in the following table represent how far off the estimates (let us call these errors) of these golfers were from the actual distance. A negative value implies that the person underestimated the distance, and a positive value implies that a person overestimated the distance. $$ \begin{array}{l|rrrrrrrrr} \hline \text { Far-sighted } & -11 & -9 & -8 & -10 & -3 & -11 & -8 & 1 & -4 \\\ \hline \text { Near-sighted } & -2 & -5 & -7 & -8 & -6 & -9 & 2 & -10 & -10 \\\ \text { No prescription } & -5 & 1 & 0 & 4 & 3 & -2 & 0 & -8 & \\ \hline \end{array} $$ At a \(1 \%\) significance level can you reject the null hypothesis that the average errors in predicting distance for all golfers of the three different vision types are the same.

Briefly explain when a one-way ANOVA procedure is used to make a test of hypothesis.
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