Exam 13: Tests of Hypotheses: Standard Deviations

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 __________.

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

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

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.

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

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 $\alpha=0.10$ .

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 $s=0.18$ .

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

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. - $s_{1}=5 \quad n_{1}=13$ $s_{2}=9 \quad n_{2}=10$ $\alpha=0.10$

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.

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.

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 $n=80$ .

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

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.

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 __________.

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

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.

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.

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