Performing a Runs Test In Exercises 15 – 20, (d) decide whethe...

Question

Performing a Runs Test In Exercises 15 – 20, (d) decide whether to reject or fail to reject the null hypothesis. Use α = 0.05

Coin Toss A coach records the results of the coin toss at the beginning of each football game for a season. The results are shown, where H represents heads and T represents tails. The coach claimed the tosses were not random. Test the coach’s claim.

H T T T H T H H T T T T H T H H

Accepted Answer

Hello everyone, and welcome back. Our next problem says a runs test yields Z approximately equal to 2.30. For a two-tailed test with alpha equals 0.05, the critical values are Z equals positive or negative, 1.96. What is your decision? A reject the null hypothesis. The sequence does not appear to be random. fail to reject the null hypothesis. The sequence appears to be random. C fail to reject the null hypothesis. There is strong evidence the sequence is not random, or D reject the null hypothesis. The test statistic is not extreme enough. So if when looking over a problem like this, we might first notice that some of our answer choices are contradictory. So, for instance, Choices B and C both say fail to reject the null hypothesis, but have opposite conclusions. B says if you fail to reject, the sequence appears to be random. C if you fail to reject strong evidence the sequence is not random. So by thinking about our null hypothesis, we can tell that some of these can be ruled out, or at least one. So recall that in a runs test for randomness, which is what we have, so it's test. For randomness. The null hypothesis is always going to be that the sequence. I random. And the alternative hypothesis will be that the sequence is not random. So, if we reject the null hypothesis, we're saying that the sequence seems to not be random. So Choice A, rejected a hypothesis sequence does not appear to be random. Well, those two things go together, that might be our answer. Choice B failed to reject the null. The sequence appears to be random. That's also consistent. But choice C failed to reject the null. There's strong evidence the sequence is not random. It would be incorrect just because failing to reject in the hypothesis would mean that there is evidence that the sequence is random, significant evidence. Then choice D reject the null hypothesis. The test statistic is not extreme enough. Well, this is also contradictory, because what we're saying when we use this test statistic is we're looking at probabilities of the results we got. Being probable if the null hypothesis is true. So whether we get those results or something more extreme, what is the probability. And so, To reject the null hypothesis, the test statistic needs to be. Extreme enough outside those critical Z values. So choice D also contradictory in its two parts. So now we just need to think about whether we're going to reject or fail to reject that null hypothesis. So, we have our Z value of 2.30, and a two-tailed test. So just a quick sketch to remind ourselves, we have a distribution. We have two Z values. Marking off two tails, so there's our. 1.96, negative Z value or lower level Z value, and our 1.96 upper level Z value. And then That region to the left of those critical Z values, that area under the curve, is going to be our rejection region. So, when our value of Z is outside of those critical Z values, we reject our hypothesis. Or in other words, when the absolute value of Z is greater than 1.96. We will reject the null hypothesis. So, we have Z approximately equal to 2.30. This is greater than 1.96, which is our critical Z value or one of our critical Z values. So, that means indeed, that we reject our null hypothesis. So our answer choice will be choice A. We reject the null hypothesis, which means the sequence does not appear to be random. So choice B is incorrect, because our Z value falls in the rejection region. So, once again, we're given a Z value. We know we have a two-tailed test, and we're given the critical Z values and asked for our decision. That would be choice A, we would reject the null hypothesis, and say that the sequence does not appear to be random. See you in the next video.

Key Concepts

Runs Test

Null Hypothesis

Significance Level (α)

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