A measure of anxiety has been designed in such a way that it is normally distributed with a mean of 0 and a standard deviation of 20. To investigate whether year-end examinations at South African universities are associated with high levels of anxiety among the student population, researchers decide to administer the anxiety questionnaire to a random sample of students.
a) If the sample size is 200, what is the standard error?
b) How large does the sample have to be to ensure that the standard error is no larger than 0.7?
c) What are the mean and variance of the sampling distribution of the mean for the anxiety questionnaire, for samples of 150 cases?
d) Knowing the mean and variance of the sampling distribution of the mean for samples of 150 cases, what is the probability that the mean will fall below an anxiety score of –2?
e) If the researcher draws a random sample of 300 students, what is the probability that the mean of this sample will be greater than 1.5?
f) If the researcher draws a random sample of 80 students, what is the probability that the mean of this sample will be less than –1?