Problem sheet 3 -- Hypothesis testing, correlation coefficients, and covariance
Hypothesis testing
A professor wants to know if her introductory statistics class has a good grasp of basic math. Six students are chosen at random from the class and given a math proficiency test. The professor wants the class to be able to score above 70 on the test. The six students get scores of 62, 92, 75, 68, 83, and 95. Can the professor have 90 percent confidence that the mean score for the class on the test would be above 70?
- Approach the problem first by doing a standard test using tables (or R functions).
- In a second approach, calculate the probability by estimating the distribution of sample means writing your own computer program.
Covariance and correlation coefficients I
Show that the covariance is linear in its arguments.
Covariance and correlation coefficients II
Consider a normally distributed random variable X with mean zero and variance sigma_X^2=5 and a normally distributed random variable Y with mean zero and variance sigma_Y^2=10. We want to consider the variables U=X+2Y and V=4X-3Y.
- Write a program that samples X and Y, calculates U and V, and estimates the covariance and correlation between U and V. Give 99% confidence intervals for your results.
- Calculate Cov[U,V] and Corr [U,V] analytically and compare to the estimates of your program.
- Would you get different results if X and Y were not drawn from normal distributions but from an arbitrary distributions with mean zero and respective variances 5 and 10?