Need the answers to Parts (a) thru (g). Please show all work so I can learn the

Question

Need the answers to Parts (a) thru (g). Please show all work so I can learn the process. Part (e) references operating curves from my text, but these curves should match your text also. Thank you very much!

2. A handful of cities in North America separated by a river... Richmond and Manchester, VA are separated by the James River Fargo, ND and Moorhead, MN are separated by the North River Minneapolis and St Paul, MN are separated by the Mississippi river... Omaha and Lincoln, NE are separated by the Platte River... and so on. These twin cities allow us to apply the dependent paired t-test So, as evil scientists, we may want to test the efficacy of a drug that grants psychic abilities. A natural (although unethical) experiment to run might work like this: go to each of these twin cities, pick one of the two sides of the river, dose your selected town with your evil chemical somehow, and then measure the intensity of the psychic field around each town as a result. We denote the psychic field over the ith (dosed) town Xi, and the (undosed) twin town across the river as Yi. We obtain the following data 0.001 0.000 0.002 0.000 0.001 0.001 0.004 0.000 0.002 0.001 (a) State and justify a null and alternative hypothesis for testing whether the serum does anything (b) Which test statistic do we use and how many degrees of freedom does it have? (c) With significance a 0.01, what is our rejection region for Ho (d) Compute the test statistic and state your conclusion for the above test (e) se the operating characteristic curves in Appendix A of the text to determine the sample size necessary to run a similar test as above with power 1-8 0.8 and with standard error d 1.0 Should we have had a larger sample to run a test with this power and standard error? What sample size would have been a better choice (f) You realize all your measurements are quite small and suspect your measurement device is producing nothing but noise. You wish to test this. Assume X1,X2, Y5 are approximately normal random variables, i i.d. with distribution N(Au, 10-6) where is unknown. Given the pop- ulations of the towns, you think rejecting Ho Au 0.003 in favor of an alternative hypothesis Ha 0.003 at significance a 0.05 (using the Z test for an uknown mean with known vari ance would be strong evidence that your measurement device is broken. Or maybe that psychics don't exist. Conduct this test, write your conclusions, and interpret your results (g) Critique your approach in the previous step: what is incorrect about the approach we used to test whether the device is working? Explain in as much (or as little detail as you feel is necessary to make your argument.

Explanation / Answer

Back-up Theory

X = Psychic field (dosed)

Y = Psychic field (undosed)

We assume: X ~ N(µ1, 12) and Y ~ N(µ2, 22) where 12 and 22 are unknown.

Now, to work out solution,

We have a sample of n (5) = observations on each of X and Y

Part (a)

Null Hypothesis/; H0: µ1 = µ2   Vs Alternative: HA: µ1 > µ2

Alternative is taken as µ1 > µ2 since it is expected that dosed ones will have a higher psychic field. ANSWER

Part (b)

Test Statistic: t = (Xbar - Ybar)/s{(2/n)}

where n = 5 = common sample size for X and Y, Xbar and Ybar are sample means of X and Y respectively and s2 = (s12 + s22)/2; s12 and s22 being the respective sample variances.

Under H0, t ~ t2n-2. So, the degrees of freedom = 8 ANSWER

Part (c)

Given = 0.01, the rejection region is ‘> t8, 0.01 i.e.,upper 1% point of t8, which, from Table of

t-distribution, is found to be 2.896.

So, the rejection region is ‘> 2.896 ANSWER

Part (d)

Computations: [given = 0.01]

Using the given data, Xbar = 0.002; Ybar = 0.0004;  

s12 = 0.0000035; s22 = 0.00000038;   s2 = 0.00000194;    s = 0.001393 and t = 1.816

From Table of t-distribution, t8, 0.01 = 2.896

Inference: H0 is accepted at 1% level of significance since t < t8, 0.01

Conclusion: Evidence is not sufficient to suggest that the serum does something.

DONE

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