Probability in statistics assignment - what is the standard

Question

X is a random variable, normally distributed (with a normal probability density function), with mean µX and variance σ₂ . Conventional shorthand notation for this is X ~ N (µX, σ² ).

X¯ is a random sample mean computed from n independent random selections of X-values form the population. Likewise for independent random variable Y , corresponding to a different population; i.e. Y ~ N (µY , σ² ).

Y¯ is also computed from the random, independent selection of n ( same n as that for the X population) individual Y-values. The population variances, σ² and σ² are known and they are not equal to each other. The population means

µ_X and µ_Y are not known.

(a) What is the standard deviation (also called standard error) of the mean X¯ , i.e. σX¯ ?

(b) What is the population variance of the difference between sample means, X¯ - Y¯ (a common notation for this variance is σ²-_X^-Y), in terms of population X and population Y parameters, and n.

(c) What is E[X¯ - Y¯ ] in terms of population parameters?

(d) Identify the probability density function (pdf) for X¯ - Y¯ using shorthand notation

like that used to identify the pdfs for X and Y ; i.e. the N ( , ) notation. (Of course, correct expressions should be used for the mean and variance of X¯ - Y¯ .

(e) A statistical test of H0 is to be done using n particular values from population X, {x₁, x₂, . . . , x_n}, and n particular values from population Y , {y₁, y₂, . . . , y_n}.The null hypothesis, H₀, is H₀ : µ_X -µ_Y ≤ µ₀. This is a one-sided test. Keep in mind that, although the null hypotheses says µX - µY ≤ µ0, the null hypothesis is tested with a test statistics in which µ_X -µ_Y is set equal to µ₀. Because the test is one-sided, deviations of the test statistic from that expected under H₀ in only one direction support the alternative hypothesis, H_A.

In this case, large positive deviations from the expected support H_A. State the alternative hypothesis, H_A.

(f) Write an equation for the conventional test statistic, giving it the name z*; lower case because it is computed from particular values of X and Y in part (e) above. Be sure to use µ0 in this test statistic, rather than µX¯ -Y¯ . µ₀ is the same as µX¯ -Y¯ only when H0 is true.

(g) Consider a population of such test statistics, each computed with n independent, random X-values, and n independent, random Y -values, so the means X¯ and Y¯ are random variables. Consequently, the test statistic is random; let's call it Z*. Write the expression
for Z* (in which µ0 should appear, as in (f) above).

(h) The test statistic Z* has different pdfs under H₀ and H_A. This is the essence of hypothesis testing. Write the expression for Z* in the case where the null hypothesis is true. In this case, the true difference of population means µ_X -µ_Y is equal to µ₀. Therefor, replace µ₀ in the test statistic in part (g) above by µ_X - µ_Y . Call this test statistic Z* to clearly

identify it as the test statistic in the case where µ_X - µ_Y = µ₀. Identify the pdf of Z* continuing to use the standard short-hand notation above).

(i) In the case where H_A is true, and µ_X - µ_Y > µ₀ by some amount δ. Consequently, µ₀ can be written as µ₀ = (µ_X - µ_Y ) - δ. Substitute this expression for µ₀ into your test statistic in part (g) above, and call it Z*_FALSE to clearly identify it as the test statistic where H₀ is false (H_A is true).

(j) This is a good time to check your work so far. Z*_TRUE should be a standard normal random variable. Express Z*_FALSE as the sum of 2 terms, where one of the terms is identical to Z*_TRUE, and the other is a constant, δ/σ _{¯X ¯Y} . If this step is not consistent with what you TRUE X-Y have obtained up to this point, there is some error somewhere.

Identify the pdf of Z*_FALSE (using the same shorthand notation).

(k) The statistical test is to be done at a significance level α, so H₀ will be rejected if Z* > z_1-α, i.e., the (1 - α) quantile of a standard normal random variable. An example is α = 0.05. We demand a power (1 - β). This means that we want a probability of rejecting H0 when it is false (so the true and hypothesized means differ by δ) to be (1 - β).

One can ask, by how much does the pdf of Z*_TRUE have to be shifted rightward to give a Z*_FALSE such that samples from the Z*_FALSE fall in the rejection region consisting of values greater than the (1 - α) quantile of a standard normal random variable, Z*_TRUE, with probability (1- β). The answer is, in order to achieve the specified power, the β quantile of Z*_FALSE must coincide with the 1 - α quantile of the standard normal random variable Z*_TRUE , z_1-α.

This shift is z_1-α - z_β.

Sketch the pdf of Z*_TRUE(call the abscissa variable z) and mark the point z_1-α on the z axis.

(l) On the same plot, sketch the pdf for Z^*_FALSE so that its β quantile coincides with the 1 - α quantile of Z*_TRUE.

(m) Write an equation for δ that gives the specified power (1 - β), at the specified value of α, with the number of sample points, n, and given σ²_X and σ²_Y .