Econ 0150 Mini-Exam 2
July 11, 2024
Instructions
You have 40 minutes to complete the exam and upload your answers to Grade-scope (30 minutes to do the exam, 10 to upload your solutions). You are allowed to use a calculator and any notes from the course. Please keep your cameras on until you upload your answers to Gradescope. Please round all answers to four decimal places unless otherwise noted (i.e. 0.9876). All questions can be solved using 1-2-3 rules. Please show your work for partial credit. Good luck!
1. Average Measurements
Suppose we take a sample of 81 individuals to find the average value of a certain measurement. Among these 81 individuals, the average value is 720 with a sample standard deviation of 270.
(a) (6 points)
Suppose we take a different sample of 81 individuals. Calculate the probability that the average value in this sample is between 660 and 720.
(b) (6 points)
Suppose we take a different sample of 81 individuals. Calculate the possible sample mean a such that P(X¯ > a) = 0.025. In other words, find the number a such that 2.5% of all sample means are greater than a.
(c) (3 points)
Explain why the normal distribution can be used to approximate the distribution of the sample mean in this case.
2. Student GPAs
Consider the GPAs of students at a university which are normally distributed with a mean of 2.4 and a standard deviation of 0.6.
(a) (6 points)
According to this model, what percentage of students should have a GPA lower than 3.0?
(b) (3 points)
If we randomly select a sample of 29 students from the university and calcu- late the average GPA, will the sample mean of GPAs be normally distributed? Explain why or why not.
3. Participation in a Survey
Assume that 40% of people living in a Pittsburgh regularly participate in a particular survey. A sample of 24 residents is taken.
(a) (3 points)
Can the normal distribution be used to approximate the distribution of the sam-ple proportion of people participating in this survey? Explain your reasoning.
(b) (6 points)
For the rest of this problem, assume the distribution of the sample proportion is normal, regardless of your answer to part (a). Calculate the probability that, in our sample, between 30% and 70% of people participate in this survey regularly.
(c) (6 points)
Now suppose we take a different sample of size 150. Calculate the probability that more than 48 people in this new sample participate in this survey regularly. Hint: the true population proportion of people participating in this survey remains the same.
3. Average Measurements
Suppose we take a sample of 81 individuals to find the average value of a certain measurement. Among these 81 individuals, the average value is 720 with a sample standard deviation of 270.
(a) (6 points)
Suppose we take a different sample of 81 individuals. Calculate the probability that the average value in this sample is between 660 and 720.
(b) (6 points)
Suppose we take a different sample of 81 individuals. Calculate the possible sample mean a such that P(X¯ > a) = 0.16. In other words, find the number a such that 16% of all sample means are greater than a.
(c) (3 points)
Explain why the normal distribution can be used to approximate the distribution of the sample mean in this case.