Question

you can calculate the p-value for a chi-square test using technology. after calculating the standardized test statistic, use the cumulative distribution function (cdf) to calculate the area under the curve. use the p-value method to test the claim.

a school administrator claims that the standard deviation for eighth-grade students on a test is greater than 35 points. a random sample of 27 eighth-grade students has a standard deviation of 36.9 points. at α = 0.01, is there enough evidence to support the administrator’s claim?

identify the null and alternative hypotheses.
a. ( h_0: sigma geq 35 )
( h_a: sigma < 35 )

b. ( h_0: sigma leq 35 )
( h_a: sigma > 35 )

c. ( h_0: sigma < 35 )
( h_a: sigma geq 35 )

d. ( h_0: sigma > 35 )
( h_a: sigma leq 35 )

identify the standardized test statistic.
( chi^2 = 28.899 ) (round to three decimal places as needed.)

identify the p-value.
( p = square ) (round to four decimal places as needed.)

Explanation:

Step1: Recall the formula for the chi - square test statistic for standard deviation

The formula for the chi - square test statistic when testing a claim about the standard deviation \(\sigma\) is \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\), where \(n\) is the sample size, \(s\) is the sample standard deviation, and \(\sigma_{0}\) is the hypothesized population standard deviation. Here, \(n = 27\), \(s=36.9\), and \(\sigma_{0}=35\). We have already calculated \(\chi^{2}=\frac{(27 - 1)\times36.9^{2}}{35^{2}}=\frac{26\times1361.61}{1225}=\frac{35401.86}{1225}\approx28.899\) (this step is just for verification of the given test statistic).

Step2: Determine the degrees of freedom

The degrees of freedom for a chi - square test for standard deviation is \(df=n - 1\). For \(n = 27\), \(df=27-1 = 26\).

Step3: Calculate the P - value

We are performing a right - tailed test (since the alternative hypothesis is \(H_{a}:\sigma>35\)). The P - value is the probability that \(\chi^{2}>\chi_{calc}^{2}\) (where \(\chi_{calc}^{2}=28.899\)) with \(df = 26\). Using a chi - square distribution calculator or software (e.g., using the chi - square CDF function), the P - value is calculated as follows:

The cumulative distribution function for chi - square is \(P(\chi^{2}\leq x)\). So, \(P - value=1 - P(\chi^{2}\leq28.899)\) with \(df = 26\).

Using a chi - square calculator (for example, in a TI - 84 Plus: 1 - chi2cdf(0,28.899,26)), we get the P - value approximately equal to 0.3675.

Answer:

0.3675