Hypothesis Testing Using Rejection Region(s) In Exercises 39–4...

Question

Hypothesis Testing Using Rejection Region(s) In Exercises 39–44, (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.

[APPLET] Gross Domestic Product A politician estimates that the mean gross domestic product (GDP) per country in a recent year is greater than $400 billion. You want to test this estimate. To do so, you determine the GDPs of 42 randomly selected countries for that year. The results (in billions of dollars) are shown in the table at the left. Assume the population standard deviation is $2099 billion. At alpha=0.06, can you support the politician’s estimate?

tab1

Accepted Answer

Hello everyone. Let's take a look at this question together. A researcher claims that the mean annual rainfall in a certain region exceeds 950 millimeters. To test this claim, you collect data from 36 randomly selected locations and find a mean rainfall of 910 millimeters. Assume the population standard deviation is 480 millimeters. At alpha equals 0.04, can you support the researcher's claim. So in order to solve this question, we have to determine whether we can support the researchers' claim that the mean annual rainfall in a certain region exceeds 950 millimeters, where we've collected data from 36 randomly selected locations and found a mean rainfall of 910 millimeters, and we assume that the population standard deviation is 480 millimeters. And based on the provided information, we should note that the sample size is and equals 36, which we can use this information to conduct our requirement check. And since the population standard deviation is known, the sample is random, and our sample size N is greater than 30, we can use the Z test. And so the first step is to state the null and alternative hypotheses. Which from the provided information we note the company claims that the mean annual rainfall in a certain region exceeds 950 millimeters, and this can be expressed as mu is greater than 950. And since it is a statement of strict inequality, it is the alternative hypothesis. So our alternative hypothesis is mu is greater than 9. 50. And the complement of the alternative hypothesis is the null hypothesis, so the null hypothesis is mu is less than or equal to 950. Therefore, our null and alternative hypotheses are a null hypothesis, which is mute is less than or equal to 950, and our alternative hypothesis, which is mute, is greater than. 950, which is the claim, and from the provided information, we know the given level of significance, which is alpha, is alpha equals 0.04. And now we can determine the critical value which is given as the subzero. And since our alternative hypothesis contains the greater than inequality symbol, the hypothesis test is a right-tailed test. So, we will use the table for standard normal distribution and look for the Z score that corresponds to a cumulative area, which is to the left of 1 minus alpha, which is equal to 1 minus 0.04, or 0.96. And since 0.96 is between 0.9599 and 0.9608, we can interpolate. And by interpolation we are given the following equation where we know that we have to solve for Z subzero, so we can rewrite this equation to solve for Z subzero, which gives us the following equation solving for Z subzero, and when simplified. The value for our critical value of Z subzero is 1.751, and using our critical value, we can determine the rejection region, and since the test is a right tail test and the critical value is 1.751, the rejection region is Z is greater than 1.751. And now we need to find the standardized test statistic Z. Using the following formula for our standardized test statistic, where Z is equal to the sample mean subtracted by the population mean, which is divided by the population standard deviation divided by the square root of our sample size N, and substituting the values into the formula, we have 910, subtracted by 950, divided by 480, which is the. Divided by the square root of 36, and simplifying this equation, our value for the standardized test statistic Z is -0.5. And then using both the critical value and the standardized test statistic, we can make our decision. And since the standardized test statistic Z is less than our critical value in a right tail test, Z is not in the rejection. And as a result, we failed to reject the null hypothesis. So based on this information, since the claim is our alternative hypothesis and we fail to reject the null hypothesis, the interpretation for this decision is given as there is insufficient evidence to support the claim that the mean annual rainfall exceeds 950 millimeters. And using this interpretation, we can identify that answer choice B is the correct answer, as there is insufficient evidence to support the claim that the mean annual rainfall exceeds 950 millimeters, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Hypothesis Testing

Rejection Region

Standardized Test Statistic (z)

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