Hypothesis Testing Using Rejection Regions In Exercises 19–26,...

Question

Hypothesis Testing Using Rejection Regions In Exercises 19–26, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic t, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.

Lead Levels As part of your work for an environmental awareness group, you want to test a claim that the mean amount of lead in the air in U.S. cities is less than 0.032 microgram per cubic meter. You find that the mean amount of lead in the air for a random sample of 56 U.S. cities is 0.021 microgram per cubic meter and the standard deviation is 0.034 microgram per cubic meter. At α=0.01, can you support the claim?

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says a university administrator claims that the average time to graduate is not greater than 4.5 years. A sample of 40 recent graduates show a mean graduation time of 4.7 years, with a standard deviation of 0.8 years. At alpha equal to 0.05, is there sufficient evidence to reject the administrator's claim? Assume the population is normally distributed. So, we need to evaluate the claim that the average time to graduate is not greater than 4.5 years. That means that the average time to graduate is going to be less than or equal to 4.5 years. The first thing we want to do is set up our hypotheses. So for our null hypothesis, H dot, we're going to have our claim here that the mean time to graduate, which will be labeled as mu, is less than or equal to 4.5 years, and our alternative hypothesis, HA is going to be that mu is greater than 4.5 years. Now, since our alternative hypothesis contains the greater than symbol, that means that the hypothesis test is going to be a right-tailed test. We're also given some other information on this problem. We're told that we have a sample of 40 recent graduates, that means our sample size end is going to be equal to 40. We're also given the sample mean, which is going to be X bar, and that was told to be 4.7 years. And we're also given the sample standard deviation S, and that is going to be equal to 0.8 years, and we're also given that alpha is equal to 0.05. Now, since the population standard deviation is unknown, the appropriate test is a one sample T test for the mean. So recall your formula for the Test statistic that is going to be T is equal to the quantity of X minus u in quantity, divided by the quantity of S divided by the square root of N in quantity. Now, we have all of those values given to us in the problem, so we can go ahead and directly calculate our T value. So that means that T is going to be equal to the quantity of 4.7 minus 4.5. In quantity, divided by the quantity of 0.8, divided by the square root of 40 in quantity. And when we evaluate this this expression, this is going to be approximately equal to. 1.581. So next step is to find our critical T value, and to do that, we'll need the degrees of freedom. Recall that our degrees of freedom, which we'll label as DF is going to be equal to N minus 1. And we already have our in value, so this is going to be equal to 40 minus 1. Which is going to be equal to 39. So we have 39 degrees of freedom. So, to find our critical value, we just need to look up on our tables. For the critical T value with 39 degrees of freedom, and alpha equal to 0.05. In doing so, we'll see that our critical Tval T sub Z is going to be equal to. 1.685. And so, when we compare our critical T value to the calculated T value that we obtain, we say that our calculated T value, T, is actually less than T sub C. So our calculated T value is less than our critical value. So that means that we fail. To reject our no hypothesis, so we failed to reject. Each night And that means that there is insufficient evidence to reject the administrator's claim. So the answer to this problem is going to be no. There is insufficient evidence to reject the administrator's claim. So that is the answer that we're looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Key Concepts

Hypothesis Testing

Rejection Region

Standardized Test Statistic

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