Question

z - scores: in the previous 2 problems you saw that the areas to the left were the same 0.2266. for elephant pregnancies the probability that a randomly selected pregnancy is less than 501 days is 0.2266. for elephant pregnancies this is written as p(x < 501)=0.2266. and for human pregnancies the probability that a randomly selected pregnancy is less than 254 days is 0.2266. for human pregnancies this is written as p(x < 254)=0.2266. in the previous 2 problems you calculated the z - score for x = 501 for elephant pregnancies and x = 254 for human pregnancies. how do those two z - scores compare against each other? the z - scores are exactly the same for x = 501 (elephant) and x = 254 (human). the z - score for x = 254 (human) is greater than the z - score for x = 501 (elephant). the z - score for x = 501 (elephant) is greater than the z - score for x = 254 (human).

Explanation:

Step1: Recall z - score property

The z - score is defined such that if \(P(X < x)=p\), the z - score \(z\) corresponding to the value \(x\) is the number of standard deviations \(x\) is from the mean, and is found using the standard normal distribution table (or z - table). If two different random variables \(X_1\) and \(X_2\) have the same cumulative probability \(P(X_1 < x_1)=P(X_2 < x_2)=p\), then the z - scores \(z_1\) and \(z_2\) corresponding to \(x_1\) and \(x_2\) respectively are equal.

Step2: Analyze given probabilities

We are given that for elephant pregnancies \(P(X < 501)=0.2266\) and for human pregnancies \(P(X < 254)=0.2266\). Since the cumulative probabilities are the same for the two different values in different distributions, the z - scores corresponding to these values are the same.

Answer:

The z - scores are exactly the same for \(X = 501\) (elephant) and \(X = 254\) (human).