n a written test given to a large class comprising 42 students, the test scores were found to be normally distributed with a mean of 78 and a standard deviation of 7. A minimum score of 60 was needed to pass the test. A score of 90 or greater was needed to earn an A in the test. Answer the following questions;
a) What is the probability that a randomly selected student would pass the test?
b) What percentage of students earned an A?
==here is what i have
== a)for stduent to pass then test P(X>60)=1-P(X<60)=1-P(Z<(60-78)/7)=1-P(Z<-2.5714)=1-0.0051=0.9949
b) for student to get A grade P(X>90)=1-P(X<90)=1-P(Z<(90-78)/7)=1-P(Z<1.7143)=1-0.9568=0.0432
total number of student who got A grade =0.0432*42=1.8144
is this correct.
I agree with it, but I find I would like to meet that .8144 of a person.
Should I round up or round down?
To round to whole people, you would round up if it was .5 of a person or greater.
a) To find the probability that a randomly selected student would pass the test, you are correct in using the formula P(X > 60) = 1 - P(X < 60). However, the calculation of P(Z < -2.5714) needs to be revised.
First, calculate the z-score for the value of 60 using the formula (x - μ) / σ.
Z = (60 - 78) / 7 = -2.5714.
Next, consult a standard normal distribution table or use a statistical calculator to find the probability associated with this z-score.
Looking up the z-score of -2.5714 in a table, you will find that the corresponding probability is approximately 0.0051 (not 0.9949 as you mentioned).
After finding the correct probability, subtract it from 1 as you did, to get the probability that a randomly selected student would pass the test.
So, P(X > 60) = 1 - 0.0051 = 0.9949.
b) To find the percentage of students who earned an A, you need to calculate the probability that a randomly selected student would score 90 or greater on the test (P(X > 90)). You followed the correct steps by calculating the z-score and finding the corresponding probability.
However, it seems like you made an error in multiplying the probability by the total number of students (42) to find the number of students who earned an A. To find the percentage, you need to divide the number of students who earned an A by the total number of students and multiply by 100.
So the revised calculation should be:
Number of students who earned an A = P(X > 90) * 42 = 0.0432 * 42 = 1.8144 (approximately).
To find the percentage of students who earned an A:
Percentage of students who earned an A = (Number of students who earned an A / Total number of students) * 100 = (1.8144 / 42) * 100 ≈ 4.32%.
So, your calculation for b) is correct, but make sure to revise the calculation for a) as explained above.