Testing the Difference Between Two Means, (a) identify the cla...

Question

Testing the Difference Between Two Means, (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 samples are random and independent, and the populations are normally distributed.

Pet Food

A pet association claims that the mean annual costs of food for dogs and cats are the same. The results for samples of the two types of pets are shown at the left. At , α=0.10 can you reject the pet association’s claim? Assume the population variances are equal. (Adapted from American Pet Products Association)

Table displaying sample statistics for annual pet food costs: dogs' mean $255, cats' mean $231, with respective standard deviations and sample sizes.

Accepted Answer

Hello everyone. Let's take a look at this question together. An automotive survey organization claims that the average annual maintenance cost for hybrid cars is the same as that for gasoline powered cars. To test this claim, the following data are collected from two independent samples. Which is hybrid cars have a sample mean of 420, a sample standard deviation of 50, and a sample size of 30, and gasoline cars have a sample mean of 445, a sample standard deviation of 55, and a sample size of 28. At the alpha equals 0.10 level of significance, can you reject the organization's claim? Is it answer choice A? No, there is insufficient evidence to reject the claim that the average annual maintenance cost for hybrid cars is the same as that for gasoline powered cars. Answer choice B, yes, there is sufficient evidence to reject the claim that the average annual maintenance. Cost for hybrid cars is the same as that for gasoline powered cars, answered Choice C. No, there is insufficient evidence to conclude the average annual maintenance cost differs for hybrid cars and gasoline powered cars or answer choice D insufficient data. So in order to solve this question, we have to recall how to test a claim so that we can test the automotive survey organization's claims that the average annual maintenance cost for hybrid cars is the same as that for gasoline-powered cars at the alpha equals 0.10 level of significance, given the following data from two independent samples, for hybrid cars and gasoline cars. And the first step in testing this claim is to state the hypotheses, which we know that the null hypothesis is mean 1 is equal to mean 2, telling us that the mean cost is the same, and our alternative hypothesis is that mean 1 does not equal mean 2, telling us the means are different. Therefore, this is a two-tailed test, and then we calculate the standard error and the test statistic, starting with the pooled variance, which the pooled variance is equal to N1 minus 1 in parentheses multiplied by the sample variance 1 plus N2 subtracted by 1 in parentheses multiplied by the. Second sample variants divided by N1 plus N2, subtracted by 2, and then plugging in the values into the pulled variance formula, we have 30 minus 1 in parentheses multiplied by 50 squad, plus 28 minus 1 in parentheses multiplied by 55 squad, all divided by 30 plus 28 subtracted by 2, and when the value for the pooled variance is equal to 2,753.125. And using the value for the pooled variance, we can calculate that our pooled standard deviation is 52.47, and then we can calculate the standard error using the standard error formula, which is equal to the pooled standard. multiplied by the square root of 1 divided by N1 plus 1 divided by N2, which plugging in the values into the formula, we have 52.47 multiplied by the square root of 1 divided by 30 plus 1 divided by 28, and when simplified, our standard error is equal to 13.787. And Then we can calculate our test statistic, which is T is equal to sample mean 1, subtracted by sample mean 2, divided by the standard error, and then substituting the values into the formula, we have 420, subtracted by 445, divided by 13.787, so our test statistic is equal to -1.813. So now we can determine our critical values and the rejection region, starting off by calculating the degrees of freedom, which the degrees of freedom is equal to N1 plus N2 subtracted by 2, so 30 + 28 subtracted by 2 is equal to 56. And so we are looking for our critical. T value at alpha equals 0.10 with a degree of freedom equaling 56. So our T critical value is approximately positive and negative 1.673. So our rejection region is that we reject the null hypothesis if the absolute value of T is Greater than 1.673. And now we can make our decision, which we know we have a calculated T statistic of -1.813, and since the absolute value of -1.813 is greater than 1.673, we reject the null hypothesis. So we can conclude that at the 0.10 significance level, there is sufficient evidence to reject the claim that the average annual maintenance cost for hybrid cars is the same as that for gasoline powered cars. And looking at our answer choices, we can identify that answer. Choice B is the correct answer. As based on our test, we know that there is sufficient evidence to reject the claim that the average annual maintenance cost for hybrid cars is the same as that for gasoline powered cars. So answer choice B is the correct answer, and all other answer choices are incorrect. I hope you found this video to be helpful. Thank you and goodbye.

Key Concepts

Hypothesis Testing

Critical Value and Rejection Region

Standardized Test Statistic (t)

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