﻿ Testing and Estimating a Single Variance or Standard Deviation Worksheet | Problems & Solutions Testing and Estimating a Single Variance or Standard Deviation Worksheet

Testing and Estimating a Single Variance or Standard Deviation Worksheet
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1.
To calculate the confidence interval for variance or standard deviation, a statistical distribution called ______ is needed. a. chi-square distribution b. binomial distribution c. poisson distribution d. multinomial distribution

Solution:

Chi-square distribution is used to calculate the confidence interval for variance or standard deviation.

Binomial distribution is used to calculate the probabilities of outcomes.

Poisson distribution is used when the sample is large and the independent variables occur over a period of time.

Multinomial distribution is the generalization of the binomial distribution. Each trial in an experiment has more than two outcomes.

2.
A chi-square variable cannot be _____, and the distributions are _____ skewed. a. positive ; negatively b. 0 ; positively c. 1 ; negatively d. negative ; positively

Solution:

A chi-square variable cannot be negative because we use the variances which are always positive and the sample n is also always positive.

The chi-square distributions are always positively skewed.

3.
At what degrees of freedom, the chi-square distribution becomes symmetric? a. 90 b. 1 c. 100 d. 50

Solution:

At 100 degrees of freedom, the chi-square distribution becomes symmetric.

4.
The area under each chi-square distribution is equal to ______. a. 0.5 or 50% b. 1.0 or 100% c. 0.9 or 90% d. 0.1 or 10%

Solution:

The area under each chi-square distribution is always equal to 1.0 or 100%.

5.
Find the values of c2(1 - α/2) and c2α/2 for a 80% confidence interval when $n$ = 4. a. 0.584, 6.251 b. 7.779, 1.064 c. 6.251, 0.584 d. 1.064, 7.779

Solution:

Degrees of freedom, d.f = n - 1 = 4 - 1 = 3

For a 80% confidence interval, α = 1 - 0.80 = 0.20

α2 = 0.202 = 0.10 and 1 - α2 = 1 - 0.10 = 0.90

With d.f = 3, c2(1 - α/2) = c20.90 = 0.584 and c2α/2 = c20.10 = 6.251
[Use the chi-square distribution table.]

6.
Find the values of c2(1 - α/2) and c2α/2 for a 99% confidence interval when $n$ = 11. a. 26.757, 2.603 b. 2.603, 26.757 c. 25.188, 2.156 d. 2.156, 25.188

Solution:

Degrees of freedom, d.f = n - 1 = 11 - 1 = 10

For a 99% confidence interval, α = 1 - 0.99 = 0.01

α2 = 0.012 = 0.005 and 1 - α2 = 1 - 0.005 = 0.995

With d.f = 10, c2(1 - α/2) = c20.995 = 2.156 and c2α/2 = c20.005 = 25.188
[Use the chi-square distribution table.]

7.
Find the values of c2(1 - α/2) and c2α/2 when α = 0.02 and $n$ = 91. a. 65.647, 118.136 b. 124.116, 61.754 c. 61.754, 124.116 d. 118.136, 65.647

Solution:

Degrees of freedom, d.f = n - 1 = 91 - 1 = 90

α = 0.02

α2 = 0.022 = 0.01 and 1 - α2 = 1 - 0.01 = 0.99

With d.f = 90, c2(1 - α/2) = c20.99 = 61.754 and c2α/2 = c20.01 = 124.116
[Use the chi-square distribution table.]

8.
The formula for the confidence interval for a variance is _________ .  a. IV. b. II. c. III. d. I.

Solution: The formula for the confidence interval for a variance is

9.
The formula for the confidence interval for a standard deviation is ________ .  a. III b. II c. IV d. I

Solution: The formula for the confidence interval for a standard deviation is

10.
A random sample of the assets (in millions of dollars) of 10 families of a colony is shown. Find the 80% confidence interval for the variance of the assets. Assume that the variable is normally distributed.
 0.5 1 0.4 1.3 0.8 1.2 0.9 2 0.7 1.5 a. between 0.158 and 0.529 b. between 0.147 and 0.518 c. between 0.179 and 0.618 d. between 0.137 and 0.437

Solution:

The problem is to find 80% confidence interval for the variance of the assets of the families (σ2).

Sample variance, s2 =  Σ(x -X)2n-1, where X is the sample mean and n, the sample size.

n = 10, X = 1.03 and s2 = 0.24
[Use calculator.]

Degrees of freedom, d.f = n - 1 = 10 - 1 = 9

For a 80% confidence interval, α = 1 - 0.80 = 0.20

α2 = 0.202 = 0.10 and 1 - α2 = 1 - 0.10 = 0.90

With d.f = 9, at α = 0.20 the chi-square critical values are c2(1 - α/2) = 4.168 and c2α/2 = 14.684
[Use the chi-square distribution table.] The 80% confidence interval for variance is

0.147 < σ2 < 0.518
[Substitute and simplify.]

Therefore, we can be 80% confident that the true variance for the assets of the families is between 0.147 and 0.518.