What is the sampling distribution of the sample mean

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Reference no: EM132257847

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Question 1 - In this question we will evaluate type I and type II error probabilities for one-sided tests. We will consider normally distributed data, with unit variance and independent observations. We will use H₀: μ = 0 for the null and H₁: μ =1 for the alternative, unless otherwise stated.

(a) Suppose we have n = 6 observations x₁, . . . , x₆. What is the sampling distribution of the sample mean (that is, of x^- = 1/6(x₁ + · · · + x₆)?)

(b) We want a test with size α = 0.05. This test is to be of the form "reject H₀ if the sample mean x^- exceeds T" (where T is a value to be determined).

You will recall that α is the probability of rejecting H₀ when true. Find an appropriate value of T.

(c) Calculate β, the probability of failing to reject the null hypothesis when the alternative is true, and state the power of the test.

(d) Consider a test of size α = 0.01. Calculate the power of this test.

(e) How many observations would it take to have a size of at most 0.01 and a power of at least 0.99?

(f) Now we will consider the case where the null and alternative hypotheses are very close. We will have H₀: μ = 0 but now H₁: μ = 0.02. Now how many observations are needed to ensure α is at most 0.01 and the power is at least 0.99?

Question 2 - Consider the following dataset:

fuel <- c(0.95, 0.52, 0.82, 0.89, 0.81)

The numbers correspond to the amount of fuel burnt by a new type of high-efficiency engine under a randomised test load. A value of 1 corresponds to the same fuel efficiency as the old engine, values greater than one correspond to more fuel burned (hence lower efficiency) and values less than one correspond to greater efficiency.

(a) One-sided or two-sided test? Justify.

(b) State a sensible null hypothesis (care!).

(d) Interpret the -Inf in the confidence interval reported by R in such a way that a nonstatistics could understand it.

Question 3 - Here we consider the amount of data needed to perform hypothesis testing.

(a) Suppose we are testing a coin using observations of tosses. We wish to test H₀: p = 0.5 against an alternative of H_A: p = 0.6 (in this question use one-sided tests only). How many tosses are needed to guarantee a size α ≤ 0.05 and β ≤ 0.2?

(b) Now generalize to consider H_A: p = 0.5 + δ. Choose sensible values for 6 and quantify the number of observations needed as a function of δ.

Reference no: EM132257847

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