Q.57 - The Practice of Statistics for AP

Question

Software gives test statistic ${\chi }^2=69.8$ and $P$-value close to 0 . The correct interpretation of this result is

(a) the probability of getting a random sample of $4877$ teens that yields a value of ${\chi }^2$ of $69.8$ or larger is basically $0$.

(b) the probability of getting a random sample of $4877$ teens that yields a value of ${\chi }^2$ of $69.8$ or larger if $H_0$ is true is basically $0$.

(c) the probability of making a Type I error is basically $0$.

(d) the probability of making a Type II error is basically $0$.

(e) it's very unlikely that these data are true.

Step-by-Step Solution

Answer

The correct answer is Option (b). 

Step 1: Given Information

Software gives test statistic ${\chi }^2=69.8$ and $P$ value close to $0$.

Step 2: Explanation

The p-value is the probability of obtaining a value (statistic) of the test result as extreme as, or more extreme than, the value that has been obtained, assuming ${\mathrm{H}}_0$ to be true. So, the p-value is not considered in making the probability. It is just a probability of getting a certain result when ${\mathrm{H}}_0$ is true.

In the given scenario, the interpretation of the result is:

The probability of obtaining a test statistic value $X^2=69.8$at least as extreme as the observed value, when the null hypothesis is assumed to be true is$0$.

That is, the probability of getting a random sample of $4877$ teens that yields a value of $x^2$ of $69.8$ or larger if${\mathrm{H}}_0$is true is basically true.

Therefore, the correct answer is Option (b).