Probability

Question

The average score of 100 students taking a statistics final was 70 with a standard deviation of 7. Assuming a normal distribution, what is the probability that a student scored greater than 65?

1. Calculate the percentage under the curve for 70 plus
2. Calculate the z value for the area under the curve for 65-70
3. Determine the percentage associated with the Z value for 65-70
4. Add the percentages for 65-70 and 70 plus

2- The average score of 100 students taking a statistics final was 67 with a standard deviation of 8. Assuming a normal distribution, what is the probability that a student scored greater than 75? Reasoning your answer out using the empirical rule.

1. Calculate the percentage under the curve for 67 plus
2. Calculate the z value for the area under the curve for 67-75
3. Determine the percentage associated with the Z value for 67-75
4. Add the percentages for 67-75 and 75 plus

 

Answer 1

(1) μ=70, σ=7.

(1.1) Since the mean is 70 the areas under the normal curve above and below 70 are the same: 50%.

(1.2) If X=65, Z=(65-70)/7=-5/7=-0.7143; if X=70 Z=0. Probabilities are respectively 23.75% and 50%, so the probability of a score between 65 and 70 is 26.25% (area under the curve). This is equivalent to a Z-score of -0.6356, equivalent to a student score of 65.55.

(1.3) 26.25% approx.

(1.4) 50+26.25=76.25%.

(2) μ=67, σ=8.

(2.1) 50%

(2.2) Z=(75-67)/8=1 for a student score of 75, a probability of 84% (for less than 75) which is 84-50=0.34=34% above the mean. Z=1 means one standard deviation above the mean.

(2.3) 34%.

(2.4) Percentage for 75+ is 16%; 34+16=50%.