Question

an insurance agent says the standard deviation of the total hospital charges for patients involved in a crash in which the vehicle struck a construction barricade is less than $3800. a random sample of 20 total hospital charges for patients involved in this type of crash has a standard deviation of $4200. at α = 0.05 can you support the agent’s claim? use the p - value method to test the claim. identify the null and alternative hypotheses. let σ be the population standard deviation. a. $h_0: σ < $3800, $h_a: σ ≥ $3800; b. $h_0: σ > $3800, $h_a: σ ≤ $3800; c. $h_0: σ ≤ $3800, $h_a: σ > $3800; d. $h_0: σ ≥ $3800, $h_a: σ < $3800. identify the standardized test statistic. (round to two 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 \(\chi^{2}=\frac{(n - 1)s^{2}}{\sigma^{2}}\) when testing a claim about the population standard deviation \(\sigma\), where \(n\) is the sample size, \(s\) is the sample standard deviation, and \(\sigma\) is the hypothesized population standard deviation.

Step2: Identify the values

We are given that \(n = 20\), \(s=\$4200\), and \(\sigma=\$3800\).

Step3: Substitute the values into the formula

First, calculate \((n - 1)\): \(n-1=20 - 1=19\).
Then, calculate \(s^{2}\): \(s^{2}=(4200)^{2}=17640000\).
And \(\sigma^{2}=(3800)^{2}=14440000\).

Now, substitute these values into the formula for \(\chi^{2}\):
\(\chi^{2}=\frac{(20 - 1)\times(4200)^{2}}{(3800)^{2}}=\frac{19\times17640000}{14440000}\)

First, calculate the numerator: \(19\times17640000 = 19\times1.764\times10^{7}=33.516\times10^{7}=3.3516\times10^{8}\)
The denominator is \(1.444\times10^{7}\)

\(\chi^{2}=\frac{3.3516\times10^{8}}{1.444\times10^{7}}=\frac{33.516}{1.444}\approx22.95\)

Answer:

22.95