In Exercises 27 and 28, (a) identify the claim and state H0 an...

Question

In Exercises 27 and 28, (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 z, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

A substance abuse counselor claims that the mean annual drug overdose death rate for the 50 states is at least 25 deaths per 100,000 people. In a random sample of 30 states, the mean annual drug overdose rate is 22.48 per 100,000 people. Assume the population standard deviation is 10.69 deaths per 100,000. At α=0.01, is there enough evidence to reject the claim?

Accepted Answer

Hi everybody and welcome back. Here's our next problem. A nutritionist claims that the average daily sodium intake for adults in a certain country is at least 3200 mg. In a random sample of 40 adults, the mean daily sodium intake is 3050 mg. Assume the population standard deviation is 400 mg. At alpha equals 0.05, is there enough evidence to reject the nutritionists claim? And then we have our answers expressed as three choices. There is sufficient evidence at the 0.05 level to reject the nutritionist claim, B being there is insufficient evidence to reject, and then C, not enough information. So, we're testing hypothesis, and we'll start with step one. That is identifying our null hypothesis and our our alternative hypothesis. So are null hypothesis H sub zero. Would be So you're gonna put a call in there instead of an equal sign. is about the mean, so we have a population mean, so we're looking at moo. And the nutritionist is claiming that the average intake is at least 3200 mg. So we'll say that mu is Greater than or equal to 3200 mg. And that means that our alternative hypothesis, not step 2, that's part two of this. So H subalpha would be That mu is less than 3200 mg. So now we'll move on to step 2 of calculating our test test statistic. So, the parameter we have is move the mean, and we know our sigma. Just highlight that briefly. In blue, so our sigma. Is 400 mg, population standard deviation. So sigma is known. And that means the test statistic we're going to calculate is Z. So, we'll write our equation for that. So we have Z equals. And then X-bar minus mu in the numerator. And sigma divided by square root of N in the denominator. And we have these values. So X bar, the sample mean, we were given that is the mean daily sodium intake in our sample of 40 adults is 3050 mg. So say that is X bar, our sample mean. So 3000. 50 And we have that minus our value. So that's our 3200 mg that were Testing And then we know that sigma is 400. And that will be divided by the square root of N, and our N here is 40, since we have a sample of 40 adults. So I'll give you a moment so you can do that math. And that should give you 2.372 as your Z value. And now for our next step, we would often get the p-value looking at our graph, and I will draw the graph just briefly so we can visualize it, even though it's a little more straightforward in this case. So we have our X-axis and our mean of 3200. And then draw our normal distribution. And we know we have this negative Z value. It's quite large. So here's our Z value of -2.3 2.372. And so the P value we're looking for. To be outside of this. And since we're looking for P to be less than alpha to be outside of this probability, then we're looking at a left-tailed test. So we then know that if our alpha, I put this on step 3, alpha is 0.05, that's given in our problem, and so that critical Z value at that alpha level, so Z sub C is -1.645. And so what is that lapel test and that says that we reject. The null hypothesis. If R Z is less than. That critical Z value. So now we're to step 4, our test conclusion. So in terms of our numerical results, we would say, since Our Z value -2.372. is less than -1.645. Then we reject the null hypothesis, and what is the null hypothesis? That's the nutritionists claim that sodium intake, or daily sodium intake is at least 3200 mg. So, we would say that our answer here then is. Choice A, there is sufficient evidence at the 0.05 level, to reject the nutritionist claim. So, once again, we went through our steps to test this hypothesis, looking at our null hypothesis and alternative hypothesis. We were given the average, we were given the mu value. Our sigma was known, so we calculated the Z value and then compared that Z value to the critical Z value for our alpha equaling 0.05. And we did a left-tailed test, finding that Z was less than the critical Z value, and therefore we could reject our hypothesis, and we can say there is enough evidence to reject the nutritionists claim that the daily average intake of sodium for adults in this particular country is Greater than or equal to 3200 mg. See you in the next video.

Key Concepts

Hypothesis Testing

Critical Value and Rejection Region

Standardized Test Statistic (z)

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