Deck 13: Tests of Hypotheses: Standard Deviations

Question 1

The symbol $X^{2}$ has which of the following properties?

A) It equals $\mathrm{X}^{2} \alpha$ .
B) The area to its right equals $\alpha$ .
C) It has $n-1$ degrees of freedom.
D) The area to its left equals $\alpha / 2$ .

Answer 1

It has $n-1$ degrees of freedom.

Question 2

To find a $90 \%$ confidence interval for the standard deviation from a small sample, we use the values

A) $X_{0.90}^{2}$ and $X_{0.10}^{2}$ .
B) $X_{0.95}^{2}$ and $X_{0.05}^{2}$ .
C) $X_{0.975}^{2}$ and $X_{0.025}^{2}$ .
D) only $X_{0.90}^{2}$ .

Answer 2

$X_{0.95}^{2}$ and $X_{0.05}^{2}$ .

Question 3

If a significance level of 0.01 is used in testing the null hypothesis that $\sigma=0.30$ against the alternative that $\sigma>0.30$ , based on a random sample of size $n=15$ , then the relevant tabled value is

A) $X_{0.01}^{2}$ .
B) $X_{0.99}^{2}$ .
C) $X_{0.005}^{2}$ .
D) $X_{0.90}^{2}$ .

Answer 3

$X_{0.01}^{2}$ .

Question 4

To test the hypothesis that two population standard deviations are equal, we use which of the following distributions?

A) $F$
B) chi-square
C) normal
D) $t$

Question 5

If the hypothesis $\sigma=0.20$ is tested against the hypothesis that $\sigma \neq 0.20$ , with $\alpha=0.05$ , based on a random sample of size $n=18$ , then the appropriate tabled value(s) is(are)

A) $X_{0.05}^{2}$ only.
B) $X_{0.025}^{2}$ and $X_{0.975}^{2}$ .
C) $X_{0.10}^{2}$ only.
D) $X_{0.025}^{2}$ only.

Question 6

A computed $X^{2}$ value that is larger than the $X_{\alpha / 2}^{2}$ value allows us to reject the null hypothesis that __________ is based on a __________.

A) $\sigma=\sigma_{0}$ , small sample
B) $\sigma=\sigma_{0}$ , large sample
C) $\sigma_{1}=\sigma_{2}$ , small sample
D) $\sigma_{1}=\sigma_{2}$ , large sample

Question 7

The hypothesis that two population standard deviations are equal can be rejected at the $10 \%$ level of significance if

A) $F>F_{0.10}$
B) $F>F_{0.05}$ .
C) $F>F_{0.20}$ .
D)

Question 8

The more that the sample standard deviation $s$ differs from the hypothesized value $\sigma_{0}$ , the more likely that the null hypothesis

A) $\sigma_{1}=\sigma_{2}$ can be rejected.
B) $\sigma_{1}=\sigma_{2}$ cannot be rejected.
C) $\sigma=\sigma_{0}$ can be rejected.
D) $\sigma=\sigma_{0}$ cannot be rejected.

Question 9

In an effort to evaluate the hypothesis that $\sigma_{1}=\sigma_{2}$ , the sample variances $s_{1}^{2}=18$ and $s_{2}^{2}=54$ were obtained, based on sample sizes of $n_{1}=15$ and $n_{2}=10$ , respectively. The numerator and denominator degrees of freedom are, respectively,

A) 14 and 9 .
B) 15 and 10 .
C) 10 and 15 .
D) 9 and 15

Question 10

If $\alpha=0.10$ , the hypothesis that $\sigma_{1}=\sigma_{2}$ , the sample variances $s_{1}^{2}=18$ and $s_{2}^{2}=54$ were obtained, based on sample sizes of $n_{1}=15$ and $n_{2}=10$ respectively, the null hypothesis should be

A) rejected since $F=3$ .
B) not rejected since $F=3$ .
C) rejected since $F=0.33$ .
D) not rejected since $F=0.33$ .

Question 11

Find the critical value needed to test the null hypothesis that $\sigma=40$ against the alternative hypothesis that $\sigma>$ 40 at $\alpha=0.01$ using a sample of size $n=80$ .

Question 12

A random sample of 11 copies of a particular mechanical component has a mean length of 5 inches with a standard deviation of $\mathbf{0 . 6 0}$ .
-Find a $95 \%$ confidence interval for the population variance.

Question 13

A random sample of 11 copies of a particular mechanical component has a mean length of 5 inches with a standard deviation of $\mathbf{0 . 6 0}$ .
-Find a $90 \%$ confidence interval for the population variance.

Question 14

A random sample of 11 copies of a particular mechanical component has a mean length of 5 inches with a standard deviation of $\mathbf{0 . 6 0}$ .
-Find a $95 \%$ confidence interval for the population standard deviation.

Question 15

A vending machine company wants to limit the variation in the number of ounces of soda that their machine dispenses into each cup. Use the 0.01 level of significance to test the null hypothesis $\sigma=0.12$ ounces against the alternative hypothesis $\sigma>0.12$ ounces based on a random sample of size $n=15$ cups for which $s=0.18$ .

Question 16

A man who needs an operation would like to evaluate the variation in survival rates in different hospitals for his type of operation. He hypothesizes that the standard deviation of survival rates at all hospitals is 0.08 . Test this hypothesis against the alternative hypothesis $\sigma \neq 0.08$ if a random sample of 12 hospitals produces a standard deviation of 0.11 . Use $\alpha=0.10$ .

Question 17

Calculate the $F$ ratio for the set of data and compare it to the appropriate tabled value. Make a decision concerning the null hypothesis of equal population standard deviations.
- $s_{1}=5 \quad n_{1}=13$
$s_{2}=9 \quad n_{2}=10$
$\alpha=0.10$

Question 18

Calculate the $F$ ratio for the set of data and compare it to the appropriate tabled value. Make a decision concerning the null hypothesis of equal population standard deviations.
- $s_{1}=15 \quad n_{1}=16$
$s_{2}=7 \quad n_{2}=10$
$\alpha=0.02$

Question 19

An owner of a chain of supermarkets claims that there is greater price competition among supermarkets than among small "convenience" food stores. A sample of ground beef prices in 13 supermarkets and 10 convenience stores gives standard deviations of $\$ 0.15$ and $\$ 0.10$ , respectively. Evaluate the claim at the $10 \%$ significance level.

Question 20

In a random sample of nine gasoline stations in New York City, the prices per gallon of unleaded gas have a standard deviation of $\$ 0.08$ per gallon. In a random sample of 14 gasoline stations in Chicago, the prices per gallon have a standard deviation of $\$ 0.03$ per gallon. Use the $10 \%$ significance level to test the null hypothesis that the price per gallon of gasoline is equally variable in the two cities.

Question 21

Finding a large-sample confidence interval for $\sigma$ requires the use of the normal distribution, whereas finding a small-sample confidence interval for $\sigma$ requires the use of $t$ distribution.

Question 22

In testing the null hypothesis that a population standard deviation equals a specified constant, the value $\sigma_{0}$ is the hypothesized standard deviation.

Question 23

The procedure used to evaluate the null hypothesis $\sigma_{1}=\sigma_{2}$ is the same as the procedure used to evaluate the null hypothesis $\sigma=\sigma_{0}$ where $\sigma_{0}$ is a constant.

Question 24

The $F$ ratio can be used to evaluate the null hypothesis that the population standard deviation equals a given constant.

Question 25

It is always true that the higher the value of $\chi^{2}$ the more likely the null hypothesis that $\sigma=28$ will be rejected.

Question 26

A hypothesis test involving one population standard deviation sometimes involves the calculation of $x^{2}=\frac{(n-1) s^{2}}{\sigma^{2}}$ .

Question 27

The value of the standard deviation $\sigma_{0}$ is never obtained from a sample.

Question 28

If $\frac{s_{1}^{2}}{s_{2}^{2}}$ has a value very close to 0 , then the null hypothesis of equal population standard deviations cannot be rejected.

Question 29

An $F$ ratio is the ratio of two population standard deviations.

Question 30

In order to test the hypothesis that a population $\sigma$ is a specified constant, it is never necessary to know the mean of the sample.

Question 31

A sample estimate of $\sigma$ is symbolized by __________.

Question 32

The mean of chi-square distribution is equal to __________.

Question 33

The possible values that a chi-square distribution can assume are __________.

Question 34

If $s_{1}^{2}=30$ and $s_{2}^{2}=20$ , then the $F$ ratio we use is __________.

Question 35

In testing the null hypothesis that a population standard deviation equals a specified constant based on a small sample, the test statistic that is used is __________.

Question 36

In tests concerning the equality of two population variances, the test statistic is __________.

Question 37

In order to convert a small-sample confidence interval for the variance to a confidence interval for the standard deviation we must __________.

Question 38

When finding a confidence interval for $\sigma^{2}$ , based on small sample, the number of degrees of freedom for the chi-square distribution is __________.

Question 39

In testing for equality of two standard deviations using the $F$ statistic of $\alpha=0.02$ , the $F$ table value that must be exceeded to reject the null hypothesis is given by the symbol __________.