In: Statistics and Probability
The following scenario applies to questions 6-8: A professor
reported that final exam scores were left skewed with a mean score
of 81.9 and a standard deviation of 6.1.
A student took a random sample of 50 students and calculated the
mean score to be 80.3. What is the probability that this student
would get a mean of 80.3 or lower?
Suppose the same student took another sample, this time of size 100, and calculated the mean. What is the probability that the student would get a mean of 84 or higher?
Suppose the student is only able to sample 25 students from the class. Can he still calculate the probability of getting an average test score higher than 84? Why or why not?
Solution:
Given: a professor reported that final exam scores were left skewed with a mean score of 81.9 and a standard deviation of 6.1.
Thus Mean= and Standard Deviation =
Part a)
Sample size= n = 50 and sample mean =
We have to find the probability that this student would get a mean of 80.3 or lower .
That is:
Since sample size = n = 50 > 30 , thus we assume large sample and hence using central limit theorem, sampling distribution of sample mean is approximately Normal with mean of sample means =
and standard deviation of sample means is:
Thus find z score:
Thus
Look in z table for z = -1.8 and 0.05 and find corresponding area.
P( Z < -1.85) = 0.0322
Thus
Thus the probability that this student would get a mean of 80.3 or lower is 0.0322
Part b) Sample size = n = 100
Find the probability that the student would get a mean of 84 or higher .
Find z score
where
Thus
Thus we get:
Look in z table for z = 3.4 and 0.04 and find area.
Thus P( Z < 3.4 ) =0.9997
Thus we get:
the probability that the student would get a mean of 84 or higher is 0.0003
Part c) Suppose the student is only able to sample 25 students from the class. Can he still calculate the probability of getting an average test score higher than 84? Why or why not?
Since sample size = n = 25 < 30, thus we can not assume large sample , hence we can not find the probability of getting an average test score higher than 84.