Question

Professor Doom has applied to the town planning board to build a Villain's Lair in a suburban neighborhood. The local residents object that he will emanate mind-control rays from his new home to make his neighbors more susceptible to his mental control. However, the town planning board approve his lair and then flee the country with ten million dollars each. A local statistician quickly jumps into action, administering a "free will" test to a random sample of 11 residents. After the lair is built, he administers the test again to each of the original participants. The scores are as follows:
before 80 82 94 87 84 88 86 85 88 92 86
after 70 82 86 91 81 80 82 81 90 85 91
The test measures a "free will quotient". People under the mental domination of another would receive lower scores.
Does the statistician have evidence that Professor Doom is controlling his neighbors?

Answer 1

Answer:

Check explanation below.

Step-by-step explanation:

Hello!

The objective of this experiment is to test if doctor Doom is controlling the neighbors' minds from his new Lair. To test this a random sample of 11 neighbors was taken and a "free will" test was administrated before the Villain's Lair was constructed and again after it was finished.

This experiment type where you have a single sample and the variable is measured "before" and "after" applying a treatment (in this case, that doctor Doom has moved into town) is a classic example of a paired sample test. You have two dependent variables, X₁: "Results of the free will test before Dr. Doom Lair is constructed" and X₂: "Results of the free will test after Dr. Doom Lair is constructed", with these variables you can establish a new variable, Xd, that I'll define as " X₁- X₂", and the objective of my hypothesis test will be to know if there is any change after the Lair is done, if there is, in fact, a change then the population mean of the difference will be nonzero, symbolically: μd ≠ 0

Using the data I've calculated the summary measures for Xd:

sample mean: Xd[bar]= 3.18

sample standard deviation: Sd= 5.44

The statistic hypothesis is:

H₀: μd = 0

H₁: μd ≠ 0

α: 0.05

(There is no signification level specified, so I've chosen the most common one)

Assuming that the variable difference, Xd, has a normal distribution, and with unknown population variance, the best statistic to use is the Students-t for paired samples:

t=  Xd[bar] - μd  ~t$_{n-1}$

Sd/√n

This test is two-tailed, so you will reject at low values of the statistic or at high values of it.

$t_{n-1;\alpha/2 } = t_{10; 0.025} = -2.228$

$t_{n-1; 1-\alpha /2} = t_{10; 0.975} = 2.228$

If t ≤ -2.228 or t ≥ 2.228 then you reject the null hypothesis.

If -2.228 < t < 2.228 then you don't reject the null hypothesis.

t= Xd[bar] - μd   =   3.18 - 0   = 1.94

        Sd/√n             5.44/√11

Since the statistic value t= 1.94 then the decision is to not reject the null hypothesis.

So, with a significance level of 5% there is not enough evidence to reject the null hypothesis, this means that the population mean of the difference on the free will test results of the neighbors before and after the Villain's Lair was constructed is equal to cero. In other words, there is not enough evidence to conclude that Dr. Doom is controlling the neighbor's minds.

I hope it helps!