Testing and Estimating a Single Variance or Standard Deviation Worksheet

**Page 1**

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 |

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.

Correct answer : (1)

2.

A chi-square variable cannot be _____, and the distributions are _____ skewed.

a. | positive ; negatively | ||

b. | 0 ; positively | ||

c. | 1 ; negatively | ||

d. | negative ; positively |

The chi-square distributions are always positively skewed.

Correct answer : (4)

3.

At what degrees of freedom, the chi-square distribution becomes symmetric?

a. | 90 | ||

b. | 1 | ||

c. | 100 | ||

d. | 50 |

Correct answer : (3)

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% |

Correct answer : (2)

5.

Find the values of c^{2}_{(1 - α/2)} and c^{2}_{α/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 |

For a 80% confidence interval,

With

[Use the chi-square distribution table.]

Correct answer : (1)

6.

Find the values of c^{2}_{(1 - α/2)} and c^{2}_{α/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 |

For a 99% confidence interval,

With

[Use the chi-square distribution table.]

Correct answer : (4)

7.

Find the values of c^{2}_{(1 - α/2)} and c^{2}_{α/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 |

With

[Use the chi-square distribution table.]

Correct answer : (3)

8.

The formula for the confidence interval for a variance is _________ .

a. | IV. | ||

b. | II. | ||

c. | III. | ||

d. | I. |

The formula for the confidence interval for a variance is

Correct answer : (4)

9.

The formula for the confidence interval for a standard deviation is ________ .

a. | III | ||

b. | II | ||

c. | IV | ||

d. | I |

The formula for the confidence interval for a standard deviation is

Correct answer : (4)

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 |

Sample variance,

[Use calculator.]

Degrees of freedom,

For a 80% confidence interval,

With

[Use the chi-square distribution table.]

The 80% confidence interval for variance is

0.147 < σ

[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.

Correct answer : (2)