Question

Suppose a Normal model is useful for describing weights of steers. The mean and standard deviation of the weights of the steers from Ranch A was found to be 1200 lbs and 60 lbs respectively. The mean and standard deviation of the weights of the steers from Ranch B was found to be 1100 lbs and 40 lbs respectively. Would a steer from Ranch A which weighed 1300 lbs be larger than a steer from Ranch B which weighed 1200 lbs relatively to each respective ranch?

A. No, because the z-score for the steer from Ranch A is larger than the z-score for the steer from Ranch B.
B. Yes, because the z-score for the steer from Ranch A is larger than the z-score for the steer from Ranch B.
C. Yes, because the z-score for the steer from Ranch A is smaller than the z-score for the steer from Ranch B.
D. No, because the z-score for the steer from Ranch A is smaller than the z-score for the steer from Ranch B.

Answer 1

A steer from Ranch A weighing 1300 lbs is not larger relatively to its ranch than a steer from Ranch B weighing 1200 lbs because the steer from Ranch B has a higher z-score, indicating it's further from its mean in terms of standard deviations.

Step-by-step explanation:

The question asks whether a steer from Ranch A weighing 1300 lbs is larger relative to its population than a steer from Ranch B weighing 1200 lbs. This comparison is done using z-scores, which are calculated by subtracting the population mean from the individual value, then dividing by the population standard deviation. For the steer from Ranch A, the z-score is calculated as (1300 lbs - 1200 lbs) / 60 lbs = 100 / 60 = 1.67, and for the steer from Ranch B, the z-score is (1200 lbs - 1100 lbs) / 40 lbs = 100 / 40 = 2.5. Since a z-score represents the number of standard deviations an element is from the mean, the steer from Ranch B with a z-score of 2.5 is relatively larger compared to the steer from Ranch A with a z-score of 1.67. Therefore, the correct answer is D: No, because the z-score for the steer from Ranch A is smaller than the z-score for the steer from Ranch B.