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practice_exam1

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

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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

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

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