Performing a Chi-Square Goodness-of-Fit Test

Question

In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ, (b) find the critical value and identify the rejection region, (c) find the chi-square test statistic, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim.

Coffee A researcher claims that the numbers of cups of coffee U.S. adults drink per day are distributed as shown in the figure. You randomly select 1600 U.S. adults and ask them how many cups of coffee they drink per day. The table shows the results. At α=0.05, test the researcher’s claim. (Adapted from Gallup)

Accepted Answer

All right. Hello, everyone. So this question says, a researcher claims that the preferences for 4 types of services are distributed as follows 40% prefer service A, 25% prefer service B, 15% prefer service C, and 20% preferer D. You survey 500 people and the results are listed in the following table. At alpha equals 0.05, test the researcher's claim using a chi square goodness of fit test. And here we have 4 different answer choices labeled A through D. So first and foremost, what are the hypotheses that we are dealing with for this scenario? First, the null hypothesis, that's Hno would state that the actual distribution matches the percentages that were claimed. Whereas the alternative hypothesis would instead state that the actual distribution does not match. The claimed percentages. So, for the chi squared test, we first have to compute the expected values that we're now going to compare or eventually going to compare to the observed frequencies given in the table. So, for that, I'm going to have to showcase the percentages that were listed in the beginning. Because recall that the. Expected frequency for each category is equal to each percentage expressed as a decimal multiplied by the sample size. So for service A, for example, E is equal to 0.40, multiplied by 500, which equals 200. For service B, that's 0.25 multiplied by 500. Which equals 125. For service C, that's 0.15 multiplied by 500, which equals 75. And for service D, that's 0.20 multiplied by 500, which equals 100. So now Recall the chi squared statistic. Is equal to the sum of. The difference between the observed and expected frequencies of each category squared and divided by the expected frequency. So here's how you're going to do this for each category, rather for each service. Let's start, for example, with service A that has an observed frequency of 220. You would take the observed frequency and subtract the expected frequency, which for service A is 200. This gives you a difference. Of 20 in this case. Once you find the difference between the observed and expected frequency, you're going to square the difference and then divide it by the expected frequency. So for service A. That would be 20 squad divided by 200, which equals 2. And so you're going to repeat this process for the remaining 3 categories. Or service B that comes out to 1.8. For service C, it's approximately 0.33. And for service D, it is 0. So now now you've found. The value for. O subtracted by E 2 divided by E for each category, you're going to add them together to find your chi square test statistic. And after adding them together, that equals 4.13. And now that you've obtained the critical value, we now have to find. Excuse me, now that we've found the test statistic rather, we find the critical value and make a comparison. But first, we have to know the degrees of freedom. Which is equal to the number of categories, just 4, subtracted by 1, which gives you 3. Using a reference. Add alpha equals 0.05 and add 3 degrees of freedom, your critical chi square value. Is 7.815. And now you compare. Because our test statistic, 4.13, is less than the critical value 7.815, we fail to reject the null hypothesis. This tells you that there is insufficient evidence to reject the claim that the distribution matches the claim. And this corresponds to option being in the multiple choice, making that your correct answer. And there you have it. So with that being said, thank you so very much for watching, and I hope you found this helpful.

Key Concepts

Chi-Square Goodness-of-Fit Test

Null and Alternative Hypotheses (H₀ and Hₐ)

Critical Value and Rejection Region

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