practice_exam1 - Practice exercises for exam 1 If you are...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Practice exercises for exam 1 If you are asked for a probability that cannot be calculated by hand, you can express your answer either in terms of normal probabilities (e.g. P ( Z < . .. ), P ( Z > . .. )), or in terms of R commands (e.g. pnorm(. ..) ). 1. Suppose our goal is to estimate the expected value μ of a population, and to provide a 95% confidence interval around our estimate. The standard deviation of the population is 2. If we aim to have a confidence interval that is around 0.25 units wide, what sample size is required? 2. Suppose we have a test statistic T that is standardized under the null hypothesis. We observe a test statistic value of T = 2 . 2. What is the two-sided p-value for our data? 3. Suppose we observe data X 1 ,...,X n , and use it to form a 95% confidence interval for the expected value of the population. Rather than using the sample standard deviation to form the interval, we use the “nominal value” σ = 1 . 5 that is based on previously collected data of a similar type. However the truth is that σ = 2. What is the actual coverage probability of our confidence interval? 4. Suppose we observe X 1 ,...,X n from a population with mean μ and standard deviation 1 / 2. For every pair of distinct observations X i ,X j , the correlation coefficient between the observations is cor( X i ,X j ) = 0 . 4. (a) What is the covariance between each pair of distinct observations? (b) What is the variance of the sample mean of these data? (c) Now suppose we are able to obtain an independent sample from a population with the same mean and standard deviation as this one. What is the variance of the sample mean in this case? 5. Suppose we are studying a quantity X that follows a normal population with mean μ and variance 1. However, we are sampling in such a way that negative values can never be included in the sample (apart from this, the sample is representative of the
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/06/2012 for the course STAT 403 taught by Professor Kerbyshedden during the Winter '12 term at University of Michigan-Dearborn.

Page1 / 4

practice_exam1 - Practice exercises for exam 1 If you are...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online