In Exercises 13 and 14, (b) find the critical values,
Us...

Question

In Exercises 13 and 14, (b) find the critical values,

Use

[APPLET] A highway patrol officer stops speeding vehicles on an interstate highway. The genders of the last 25 drivers who were stopped are shown, where F represents a female driver and M represents a male driver. Can you conclude that the stops were not random by gender?

F M M M F M F M F F F M M

F F F M M M F M M F F M

Accepted Answer

Hi everybody, let's look at our next problem. A researcher suspects that coin flips from a vending machine are not fair. In 20 flips there are only 5 heads. Use a two-tailed binomial test with alpha equals 0.05. Let N equals 20, P equals 0.5, and use binomial critical values. What are the critical values for this test? A 10 and 20, B 5 and 10, C, 4 and 16, or D, 5 and 15? So we always start by defining our hypothesis when we're testing something like this. So, we are looking to see whether coin flips are fair. So we would say that the default situation would be that they are fair. And if they are, the probability. Of we'll say capital X, where we're going to define that capital X as the number of heads. In a sample or a population, that probability of X is going to equals 0.5, if the flips are fair. Every time you flip the coin, you have a 50% chance of getting hits. So that means our alternate hypothesis would be that they are not fair. Which would just mean that the probability of X is not equal to 0.5. It's not a less than or greater than anything that's not equal to 0.5. If we have more heads than we expect or fewer heads than we expect, that is going to meet that condition. So now we're going to think about using a binomial distribution, a binomial test to find those critical points. So, we're told in the problem, it's a two-tailed test, but we can see that by the fact that we have this um does not equal sign. So let's put two-tailed test. were given that alpha equals 0.05, but recall that in a two-tailed test, that would be the area of both tails. And so, What we need to look for to determine our critical values is, so put two tails. And that means that we need to look at. Alpha divided by 2. And that's going to equal 0.025, as the area of each individual tail. So that area needs to be greater than or equal to. The probability of. Capital X of getting heads. To be able to reject our hypothesis. The probability of what we obtain needs to be, you know, that that result needs to be so unusual that we can reject the null hypothesis. So we know it's very straightforward in this example, that we're going to have an expected value or mean, so mu. Equal to just our number of samples, so 20, number of Flips, I should say not number of samples. Number of flips in the sample is 20 multiplied by the probability. 0.5, same view of X, and that's just going to be 10. Obviously, you could do that in your head. So, that's our expected value, and what we're looking for is the values on either side of that, that are so small or so large that the probability of getting them will be. Less than that, 0.05, or the probability in each tail will be. Less than 0.025. So, we need to look for the smallest X, in this case it's a small lower case x, where if our capital X, if our number of heads is less than or equal to that X, The probability of X, it's not equals there. I need A is less than or equal to 0.025. Now, we have this symmetry, we're looking at two tails, so we also need the largest X. If our capital X is greater than or equal to that X value, that P X is less than or equal to 0.025. That would be our two tails. So, can you use your binomial table or a calculator. In my case, I looked up on the binomial table, and I'll just mention what you would do there. So, at on the binomial table, you're going to take that probability of 0.025 as your P. When n is 20 and look for the x value that has. A probability that is less than 0.025, or we're starting with the smallest X value. And when I looked at my table, I found that the probability of getting 5 heads was 0.015, while the probability of getting 6 heads was 0.036. So 6 is going to be too large. It will not be out in that tale. And getting 6 heads would not give you enough evidence to reject your hypothesis. So our lower critical value. is going to be 5. Well, we could then go and look up that 0.025 on the numbers higher than 10 are mean, but remember that this distribution is symmetrical. So, if our lowest X is 5, Which is 10 minus 5, then by symmetry, we will find the same. Add a value of 5 greater than our means. So by symmetry. Our larger X, so our other critical value will be. X X greater than or equal to 15. And indeed, if you looked that up on your binomial table, that would give you that same 0.0154%. So we have our two critical values, and We don't actually need to go any farther into whether we reject the null hypothesis or not, but let's recall that meaning of those critical values. Is that we would reject the null hypothesis. If our number of heads or X is less than or equal to 5, Or if our number of heads, X is greater than or equal to 15. So, when we look at our answer choices, we do have this choice D, which is 5 and 15. So choice D is our answer. And just a quick note that if you were kind of running out of time on a test, you wanted to narrow down your choices, that we would recall, since we know the mean here, that our critical values must be symmetrically distributed on either side of the mean. So, A and B are not possible because they are not symmetrical. Since we have that two-tailed test, and then choice C 4 and 16. That's going to be too stringent because we have when X is equal to 5 or X is equal to 15, that gives us a probability of less than our alpha divided by 2, less than that 0.025. So we'd say this is too stringent. So once again, in our scenario, where researchers suspecting that coin flips from a vending machine are not fair, getting only 5 heads and 20 flips. So, they're going to use a two-tailed binomial test with an alpha of 0.05. We know that N will equal 20, P will equal 0.5, and we're going to use binomial critical values, and the question's asking us, what are the critical values for this test? And the answer is choice D 5 and 15. So, just for fun, if we look at it, that means if only 5 heads appeared in 20 flips, then the researcher has enough evidence to reject that hypothesis and suspected that vending machine is not fair. See you in the next video.

Key Concepts

Critical Values

Hypothesis Testing

Chi-Square Test

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