Are Seat Belts Effective? A simple random sample of front-seat...

Question

Are Seat Belts Effective? A simple random sample of front-seat occupants involved in car crashes is obtained. Among 2823 occupants not wearing seat belts, 31 were killed. Among 7765 occupants wearing seat belts, 16 were killed (based on data from “Who Wants Airbags?” by Meyer and Finney, Chance, Vol. 18, No. 2). We want to use a 0.05 significance level to test the claim that seat belts are effective in reducing fatalities.

b. Test the claim by constructing an appropriate confidence interval.

b. Test the claim by constructing an appropriate confidence interval.

Accepted Answer

Welcome back, everyone. A study investigates the effectiveness of a new safety helmet for a cyclist. Out of 1800 cyclists not wearing the new helmet, 27 suffered serious head injuries. Out of 5400 cyclists wearing the new helmet, 9 suffered serious head injuries. At the 0.05 significance level, test the claim by constructing an appropriate confidence interval that the new helmet reduces the risk of serious head injury. Now, how are we going to how are we going to test this claim using the an appropriate confidence interval? OK, how can we construct this interval? Well, first, let's define the proportions. Now, from the data, OK, for those non-helmeted cyclists, OK, if we can refer to that group as group one, OK. Then the number of non-helmeted cyclists who have received serious head injuries is 27, and that's out of 1800 cyclists, so the sample size is 1800, and that means that the sample proportion, OK, for non-helmeted cyclists. OK, a sample proportion. Is going to be equal to 0. All right, 27 divided by 1800, which equals 0.015. On the other hand, OK, for the helmeted cyclists, if we refer to them as group 2. Then 9 of them at X2 receive serious head injuries, and this is out of 5400 cyclists, so the sample size is 5400, which means that its sample proportion or their sample proportion will be 9 divided by 5400, which equals 0.00167. Now why is the sample proportion so important? Well, recall that for the confidence interval. The bones, OK. Bones of the confidence interval are going to be equal to the difference between the proportions, OK. Plus R minus Z or critical Z value multiplied by the standard error. So for the lower bound, it's the difference minus Z multiplied by SE, and for the upper bound it's the difference plus Z multiplied by SE, OK. So now we need to figure out our standard error and our critical Z value to then figure out the confidence interval. Now here to compute the standard error of difference in proportions we can use the formula. That says the standard error. Let me put it in red just to show that we've started some work. The standard error is gonna be equal to the square root, OK, of P1 multiplied by 1 minus P1 divided by N1 plus P2 multiplied by 1 minus P2 divided by N2. Now what's that equal to? But, here, if we substitute those values, OK, then it's gonna be 0.015 multiplied by 1 minus 0.015, divided by 1800. Plus 0.00167 multiplied by 1 minus 0.00167, OK, divided by 5400. Now if we go ahead and substitute some values here, OK, then if we calculate in our numerators, that's gonna be the square root of 0.014775 divided by 1800 plus, OK, 0.001668 divided by 5400. Which when we go ahead and solve here, then we should get this to be equal to the square root of 8.54 multiplied by 10 to -6, which is equal or approximately equal. To 0.00293. That's our standard error. Know that we have our standard error. Next, let's calculate the critical Z value. Well, here, well, before we do that, actually, let's find a difference in proportions, OK? And the difference in proportion is gonna be P1 minus P2. OK, which is equal to 0.015 minus 0.00167, which equals 0.01333. This will just make it a bit easier before we continue to figure out the bones of the confidence interval. So, let's take a quick check. What do we already have? We have the standard error, the difference in proportions between the non-helmeted and helmeted groups. Next, let's find our critical value, OK? Now here since. Our first proportion is greater than our 2nd proportion. Notice that our difference is positive. Then we can use the one-tailed test. OK, the one tailed test. At the 0.05 significance level, OK? I know, since it's a one tail test for a 0.05 significance level, that means we would use, OK, a 90%. One tail. Confidence interval. Because here if we think about it, OK, if we set up our distribution. Then if it's at a 0.05 significance level, then this side will be 5%, which means the other side, they're not interested in it, but the other side will be 5%. So thus our confidence interval will be 90%, OK? So no. In that case, OK, then our critical Z value. So that is of 0.05 will be equal to 1.645. Know that we have our standard error or difference in proportions under Z values we can construct the confidence interval. Because no, this means then for the lower bone. It's gonna be equal to 0.01333 minus 1.645 multiplied by 0.00293, OK. Here Which, OK, is gonna be approximately equal to 0.00853. But for our upper bones. It's gonna be equal to 0.01333 plus 1.645 multiplied by 0.00293. And that, OK, will be approximately equal to 0.01813. So, what does that mean? Well, no, this tells us that our confidence interval. OK, let me see if I can, uh, I don't think I have that much space left on my page, but if we make a little bit downwards here, then our confidence interval is going to be between 0.0853 to 0.01813, OK? I know what does this tell us? Well, notice that since the entire interval is above 0, we are 90% confidence that the true proportion of serious head injuries is greater without the helmet than with it. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

Confidence Interval

Proportion Comparison

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