Finding P-values
In Exercises 5–8, either use technology...

Question

Finding P-values

In Exercises 5–8, either use technology to find the P-value or use Table A-3 to find a range of values for the P-value. Based on the result, what is the final conclusion?

Cotinine in Smokers The claim is that smokers have a mean cotinine level greater than the level of 2.84 ng/mL found for nonsmokers. (Cotinine is used as a biomarker for exposure to nicotine.) The sample size is n = 902 and the test statistic is t = 56.319.

Accepted Answer

Welcome back, everyone. A car manufacturer claims the average fuel efficiency of their new model is 40 MPG. A sample test yields a T statistic of 1.6. At the 0.05 significance level, should the null hypothesis be rejected in a two-tailed test? A says we should reject the null hypothesis, and the average fuel efficiency is significantly different from 40 MPG. B says we fail to reject the null hypothesis, and there is insufficient evidence to support the claim that the fuel efficiency differs from 40 MPG. C says to reject the null hypothesis. The average fuel efficiency is significantly greater than 40 MPG, and D says to fail says we fail to reject the null hypothesis because the sample size is not sufficiently large to determine significance. Now if we're going to figure out if the non hypothesis should be rejected in a two-tailed test, we need to establish or define our hypotheses. Now notice that we are testing here the claim that the average fuel efficiency of a car manufacturer's new model is 40 MPG, so the null hypothesis. OK. Is that mu equals 40. In other words, the average fuel efficiency is 40 MPG, which makes our alternative hypothesis H1, that mu is not equal to 40. The average fuel efficiency is not 40 MPG. What do we know so far? Well, we're given a T statistic in our problem of 1.6, OK, and a significance level alpha of 0.05 for a two-tailed test. How can we know if the null hypothesis should be rejected? Well, if we picture this on a normal distribution, OK. Then what we're saying here is that for a two-tail test, we're going to have our critical T values. Let's call that TC, OK? And for our critical T values, if the T statistic falls within the region of the critical T values, then we should, we fail, OK, to reject the null hypothesis. But if it falls outside the region of our critical value, then we would reject the null hypothesis, OK? or there is enough evidence for us to reject the null hypothesis. So now we need to figure out our critical values since we already have our test statistic and see how the the the T statistic compares to those critical values. No, we already know. OK, let me put this in red. That for a two-tailed test, OK? At alpha equals 0.05, we would split the significance level into two tails, so each tail has an area of 0.025. And no, the criticality value depends on the degrees of freedom. As the degrees of freedom approaches infinity, in other words, as the sample size increases, the T distribution approaches the standard normal distribution. I know our criticality value will approach 1.960. And let me put that beside uh our statement here, OK? So what does that mean? Well, no, in a two-tail test, if we go back to our distribution, standard normal distribution, then our criticality values would be -1.96 and 1.96. And no, like we just said, our test statistic is 1.6, and that test statistic would fall inside the region between. OK, so let me actually try to put that better here. It would fall inside the region. Between uh one point between our criticality values. Sorry. So what does this tell us? Well, let's write this out here actually, OK, because what this means then. That since the T statistic. Is less since the statistic is less than the critical value, OK, we fail to reject the null hypothesis. OK, no hypothesis for all degrees of freedom. Including light samples where the t distribution approaches the normal distribution, so we can conclude that there is not enough evidence to claim the average fuel efficiency differs from 40 MPG regardless of sample size. If we go back to our answer choices, then that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

P-value

Hypothesis Testing

Test Statistic

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