Deck 5: Standardization and Z Scores

Question 1

Suppose that in the population of chickens, the average number of feathers per chicken is 1000 with a standard deviation of 200. What is the probability of randomly selecting an individual from this population that has at least 1100 feathers?

Answer 1

(1100-1000)/200 = .50. Probability = .3085.

Question 2

Suppose that on a talent show like American Idol, the only contestants who actually make it onto the show must be at least 3 standard deviations above the mean in terms of their talent. What percentage of those who audition would you expect to actually make it onto the show?

Answer 2

Three standard deviations above the mean is a z score of 3. In Appendix A, the box on the second page shows that when the z score is three, the area probability of getting a value between that z score and infinity is .001350, or .135 percent.

Question 3

Suppose that in a class with 50 students in it with an average score on an exam of 30 and a standard deviation of 10, John gets a score of 45. Calculate his z score and explain what it tells you.

Answer 3

John's z score is (45-30)/10 $\rightarrow$ 15/20 = 1.5 So John's z score is 1.5, and this tells us that John's test score is 1.5 standard deviations above the mean for his class.

Question 4

If a test has a mean of 500 and a standard deviation of 50 (and is normally distributed), what score marks the 15^th percentile?
a. On this same test, what proportion of students would be expected to receive scores between 475 and 550?
b. What scores fall at the extreme 10% of this distribution (two-tailed)?