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.

a. Test the claim using a hypothesis test.

a. Test the claim using a hypothesis test.

Accepted Answer

Welcome back, everyone. A study investigates the effectiveness of a new safety helmet for cyclists. 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, that's the claim using a hypothesis that the new helmet reduces the risk of serious head injury. Now if we're going to test this claim first, let's define our hypotheses. Now the null hypothesis assumes that there is no difference in the proportions of serious head injuries between cyclists wearing the helmet and those not wearing it. Well the alternative hypothesis claims that the helmet reduces the risk of serious injury. OK, that is, the proportion of serious injuries among helmeted cyclists is lower than the proportion among non-heelmeted cyclists. So that means then that are not hypothesis HO will be that P1 is less than or equal to P2. There's no difference in injury rates. And our alternative hypothesis, H1, is that P1 is greater than P2. The injury rate is lower for a cyclists wearing the helmet. So here we consider P1 as the proportion of cyclists suffering serious head injuries in the group, not wearing the helmet. OK, so we are P1 are those not wearing the helmet. OK. While P2 Helmut, let me spell that properly. P2 is the proportion of cyclists suffering serious head injuries in the group wearing the helmet, so P2 wears the helmet. Now how can we figure out how can we test this claim? Well, we can test or we can find our Z statistic for proportions. In other words, OK, we can reject, OK, the null hypothesis. Reject the null hypothesis. If our Z or test statistic Z is greater than the critical Z value Z C. So now we need to find first our test statistic Z and then our critical value at the 0.05 significance level for this problem. Now, let's start with our Z statistic, OK? Already, we know that in our first group, X1 is 27, OK. And N1 is 1800, that is cyclists not wearing the new helmet, while X2 equals 9 and N2 equals 5400. That is out of those that are wearing the new helmet. So from this then we can calculate our sample proportion because we need that to put in the test statistic. Because our test statistic Z is equal to P1 minus P2, the difference in sample proportions divided by the standard error, where the standard error is equal to the square root of the pooled proportions multiplied by 1 minus the pool proportions multiplied by the reciprocal of N1 plus the reciprocal of N2. So you know there's there's quite a few things that we need to calculate here. First, let's start with our proportions. Now we know that for P1, it's gonna be equal to 27 divided by 1800, which equals 0.015. Next, with helmet P2. The proportion is going to be 9 divided by 5400, which equals 0.0016667, OK? Now, in this case, that means our pulled proportion that we need for the standard error will be equal to X1 + X2 divided by N1 plus N2, which equals 27 + 9. Divided by 1800 plus 5400. That's 36 divided by 7 to 200, which equals 0.005. Thus That means our standard error is gonna be equal to the square root of 0.005. Multiplied by 1 minus 0.005 or 0.995. Multiplied by 1 divided by 1800 plus 1 divided by 5400. Which is going to be equal to when we work all of that out, approximately 0.00192. I know that we have the standard error, that means our test statistic Z will be equal to P10.015 minus P2 0.0016667. All divided OK by 0.00192. And when we work at all, we get our Z test statistic to be approximately 6.946. So now we have our Z value or Z test statistic, and we need to compare that to our critical value for the information that we are given. No. For our critical Z value, OK, in other words, for a. For alpha being equal to 0.05, OK. Alpha being equal to 0.05. Then if we use our Z table for critical values, the critical value that we get is going to be equal to 1.645. Now how do both of these compare? Well, at this point you can probably tell, OK, that the Z statistic. OK. The Z statistic is greater than the critical value. 6.946 is greater than 1.645. So what does this mean? Well, based on the rule we established earlier, we are going to reject our null hypothesis, OK? We're rejecting our null hypothesis. So let's make our statement then because we're rejecting our no hypothesis, this means that at the 0.05 significance level. Significance, let's write that properly here. There is In sufficient evidence. To support the claim. OK. That the new helmet. Reduces. The risk of serious head injuries. Or a cyclist, OK? So evidence supports the claim. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Hypothesis Testing

Significance Level

P-Value

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