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F Distribution and ANOVA: Review

Module by: Barbara Illowsky, Ph.D., Susan Dean. E-mail the authors

Summary: This module provides a review of F Distribution and ANOVA as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Note: You are viewing an old version of this document. The latest version is available here.

The next two questions refer to the following situation:

Suppose that the probability of a drought in any independent year is 20%. Out of those years in which a drought occurs, the probability of water rationing is 10%. However, in any year, the probability of water rationing is 5%.

Exercise 1

What is the probability of both a drought and water rationing occurring?

Solution

0.02

Exercise 2

Out of the years with water rationing, find the probability that there is a drought.

Solution

0.40

The next three questions refer to the following survey:

Table 1: Favorite Type of Pie by Gender
  apple pumpkin pecan
female 40 10 30
male 20 30 10

Exercise 3

Suppose that one individual is randomly chosen. Find the probability that the person’s favorite pie is apple or the person is male.

Solution

100 140 100 140 size 12{ { { size 8{"100"} } over { size 8{"140"} } } } {}

Exercise 4

Suppose that one male is randomly chosen. Find the probability his favorite pie is pecan.

Solution

10 60 10 60 size 12{ { { size 8{"10"} } over { size 8{"60"} } } } {}

Exercise 5

Conduct a hypothesis test to determine if favorite pie type and gender are independent.

Solution

p-value=0p-value=0; Reject null; Conclude dependent events

The next two questions refer to the following situation:

Let’s say that the probability that an adult watches the news at least once per week is 0.60.

Exercise 6

We randomly survey 14 people. On average, how many people do we expect to watch the news at least once per week?

Solution

8.4

Exercise 7

We randomly survey 14 people. Of interest is the number that watch the news at least once per week. State the distribution of XX. XX ~

Solution

B 14 , 0 . 60 B 14 , 0 . 60 size 12{B left ("14",0 "." "60" right )} {}

Exercise 8

The following histogram is most likely to be a result of sampling from which distribution?

Figure 1
Histogram with 7 bars.

  • A. Chi-Square
  • B. Geometric
  • C. Uniform
  • D. Binomial

Solution

D

Exercise 9

The ages of De Anza evening students is known to be normally distributed. A sample of 6 De Anza evening students reported their ages (in years) as: 28; 35; 47; 45; 30; 50. Find the probability that the average of 6 ages of randomly chosen students is less than 35 years.

The next three questions refer to the following situation:

The amount of money a customer spends in one trip to the supermarket is known to have an exponential distribution. Suppose the average amount of money a customer spends in one trip to the supermarket is $72.

Exercise 10

Find the probability that one customer spends less than $72 in one trip to the supermarket?

Solution

0.6321

Exercise 11

Suppose 5 customers pool their money. (They are poor college students.) How much money altogether would you expect the 5 customers to spend in one trip to the supermarket (in dollars)?

Solution

$360

Exercise 12

State the distribution to use is if you want to find the probability that the average amount spent by 5 customers in one trip to the supermarket is less than $60.

Solution

N 72 , 72 5 N 72 , 72 5 size 12{N left ("72", { { size 8{"72"} } over { size 8{ sqrt {5} } } } right )} {}

Exercise 13

A math exam was given to all the fifth grade children attending Country School. Two random samples of scores were taken. The null hypothesis is that the average math scores for boys and girls in fifth grade are the same. Conduct a hypothesis test.

Table 2
  n n size 12{n} {} x ¯ x ¯ size 12{ {overline {x}} } {} s 2 s 2 size 12{s rSup { size 8{2} } } {}
Boys 55 82 29
Girls 60 86 46

Solution

p-value=0.0003p-value=0.0003; Reject null; Conclude averages are not equal

Exercise 14

In a survey of 80 males, 55 had played an organized sport growing up. Of the 70 females surveyed, 25 had played an organized sport growing up. We are interested in whether the proportion for males is higher than the proportion for females. Conduct a hypothesis test.

Solution

p-value=0p-value=0; Reject null; Conclude proportion of males is higher

Exercise 15

Which of the following is preferable when designing a hypothesis test?

  • A. Maximize αα size 12{α} {} and minimize ββ size 12{β} {}
  • B. Minimize αα size 12{α} {} and maximize ββ size 12{β} {}
  • C. Maximize αα size 12{α} {} and ββ size 12{β} {}
  • D. Minimize αα size 12{α} {} and ββ size 12{β} {}

Solution

D

The next three questions refer to the following situation:

120 people were surveyed as to their favorite beverage (non-alcoholic). The results are below.

Table 3: Preferred Beverage by Age
  0 – 9 10 – 19 20 – 29 30 +   Totals
Milk 14 10 6 0 30
Soda 3 8 26 15 52
Juice 7 12 12 7 38
Totals 24 30 44 22 120

Exercise 16

Are the events of milk and 30+:

  • a. Independent events? Justify your answer.
  • b. Mutually exclusive events? Justify your answer.

Solution

  • a. No
  • b. Yes, P Mand 30 + = 0 P Mand 30 + = 0 size 12{P left ( ital "Mand""30"+{} right )=0} {}

Exercise 17

Suppose that one person is randomly chosen. Find the probability that person is 10 – 19 given that he/she prefers juice.

Solution

12 38 12 38 size 12{ { { size 8{"12"} } over { size 8{"38"} } } } {}

Exercise 18

Are Preferred Beverage and Age independent events? Conduct a hypothesis test.

Solution

No; p-value=0p-value=0

Exercise 19

Given the following histogram, which distribution is the data most likely to come from?

Figure 2
Histogram with 8 bars.

  • A. uniform
  • B. exponential
  • C. normal
  • D. chi-square

Solution

A

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