Question

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

a hospital spokesperson claims that the standard deviation of the waiting times experienced by patients in its minor emergency department is no more than 0.5 minutes. a random sample of 23 waiting times has a standard deviation 0.7 minutes. at α = 0.05, is there enough evidence to reject the spokesperson’s claim?

identify the null and alternative hypotheses.

a. h₀: σ < 0.5, hₐ: σ ≥ 0.5
b. h₀: σ ≤ 0.5, hₐ: σ > 0.5
c. h₀: σ ≥ 0.5, hₐ: σ < 0.5
d. h₀: σ > 0.5, hₐ: σ ≤ 0.5

identify the standardized test statistic.
χ² = 43.12 (round to two decimal places as needed.)

identify the p-value.
p = \boxed{} (round to four decimal places as needed.)

Explanation:

Step1: Recall the formula for the P - value in a chi - square test for variance

The test statistic for a chi - square test of variance is given by \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma_{0}^{2}}\), and the P - value is the probability of getting a test statistic as extreme as or more extreme than the observed test statistic, assuming the null hypothesis is true. For a right - tailed test (since \(H_{a}:\sigma>0.5\)), the P - value is \(P(\chi^{2}>\chi_{calc}^{2})\) where \(n\) is the sample size, \(s\) is the sample standard deviation, \(\sigma_{0}\) is the hypothesized population standard deviation, and \(\chi_{calc}^{2}\) is the calculated test statistic.

We know that \(n = 23\), so the degrees of freedom \(df=n - 1=23 - 1 = 22\), and the calculated test statistic \(\chi^{2}=43.12\).

Step2: Use the chi - square distribution to find the P - value

We need to find \(P(\chi^{2}_{22}>43.12)\). Using a chi - square distribution table or a statistical software (such as a TI - 84 Plus calculator or R), we can calculate this probability.

Using a calculator (for example, in R, we can use the command pchisq(43.12,22,lower.tail=FALSE)), or using a chi - square calculator:

The chi - square distribution with \(df = 22\). The cumulative distribution function \(F(x)=P(\chi^{2}\leq x)\). So \(P(\chi^{2}>x)=1 - F(x)\).

Looking up in a chi - square table or using a calculator, we find that \(P(\chi^{2}_{22}>43.12)\approx0.0052\) (rounded to four decimal places).

Answer:

0.0052