Deciding on a Distribution In Exercises 31 and 32, decide whet...

Question

Deciding on a Distribution In Exercises 31 and 32, decide whether you should use the standard normal sampling distribution or a t-sampling distribution to perform the hypothesis test. Justify your decision. Then use the distribution to test the claim. Write a short paragraph about the results of the test and what you can conclude about the claim.

Tuition and Fees An education publication claims that the mean in-state tuition and fees at public four-year institutions by state is more than $10,500 per year. A random sample of 30 states has a mean in-state tuition and fees at public four-year institutions of $10,931 per year. Assume the population standard deviation is $2380. At α=0.01, test the publication’s claim.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a publication claims that the mean monthly cost of a certain type of subscription is more than $25. A random sample of 30 subscriptions shows a mean cost of $26.10 and the population standard deviation is known to be $4.50. At a significance level of alpha equal to 0.01, test the claim. So, the first thing we want to do is identify our claim. And our claim here is that the mean monthly cost of a certain type of subscription, which we'll label as me, is going to be greater than $25 so me is going to be greater than $25. Now, since our claim here has a strict inequality, since we have just greater than, that means we're going to use our claim as our alternative hypothesis. So, our alternative hypothesis, HA It is going to be that you is greater than 25. And our null hypothesis, H0 is going to be the complement of this statement. So H0 is going to be mu is going to be less than or equal to 25, that is the complement of mu greater than 25. And notice here that in our little hypothesis, we have an equality, because we have less than or equal to in our statement. Now, since our alternative hypothesis contains the greater than sign in it, that means that our hypothesis test is going to be right-tailed. We're also given several other pieces of information in our problem. We're given our sample size in, which is going to be equal to 30, since we have a sample of 30 subscriptions. We're also given the sample mean, which is going to be X bar, and that is going to be equal to the $26.10. We're also given the population standard deviation, which is going to be denoted as sigma, and that's going to be equal to the $4.50. And then finally, we're given our significance level of alpha, which is equal to 0.01. So now what we want to do is to test our claim. Now, since the population standard deviation is known, the appropriate test is going to be a one sample Z test for the mean. So, we call your formula for the test statistics Z. And recall that that formula is going to be Z is equal to the quantity of X minus U in quantity, divided by the quantity of sigma divided by the square root of N in quantity. And we have values for each of those quantities, so we can substitute in those values. So that means that Z is going to be equal to the quantity of 26.10. Minus 25 in quantity, divided by the quantity of 4.50. Divided by the square root of 30 in quantity. And when we evaluate this expression, this is going to be approximately equal to. 1.339. So next we need to actually find our critical Z value. So, we need to find the critical value for a one-tailed Z test at alpha equal to 0.01. And so we're just going to use our standard normal distribution table. And we see that our critical values, Z sub C is going to be equal to. 2.326. Now, since our test here is going to be right-tailed, that means that our rejection region is going to be values of Z that are greater than our critical Z value here. However, if we compare our critical value to our calculated Z value, we see that our calculated Z value, Z is actually less than our critical Z value Z sub C. So it falls outside of the rejection region. So that means that we are going to fail. To reject our hypothesis. So we're going to fail to reject each knot. And by failing to reject our no hypothesis, that means that there there is going to be insufficient evidence. To support our claim. So there is insufficient evidence to support the claim that the mean monthly cost of a certain type of subscription is more than $25. So that there 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

Standard Normal Distribution vs. t-Distribution

Hypothesis Testing

P-Value and Conclusion

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