Does the Treatment Affect Success? The following table lists f...

Question

Does the Treatment Affect Success? The following table lists frequencies of successes and failures for different treatments used for a stress fracture in a foot bone (based on data from “Surgery Unfounded for Tarsal Navicular Stress Fracture,” by Bruce Jancin, Internal Medicine News, Vol. 42, No. 14). Use a 0.05 significance level to test the claim that success of the treatment is independent of the type of treatment. What does the result indicate about the increasing trend to use surgery?

Accepted Answer

All right, hello, everyone. So, this question says the table below shares a number of students who passed or failed a certification exam based on the method of study they used. At the 0.05 significance level, test the claim that the exam result that is pass fail, is independent of the study method. What does the result indicate about the differences, or excuse me, the effectiveness of structured courses compared to self-study? So first, let's begin by detailing the new hypothesis and the alternative. Right, so the null hypothesis or h knot. is simply that the exam result is independent of the study method that students used. And so the alternative hypothesis or H1 is the opposite, right? Instead, the alternative hypothesis would state that the exam result is not independent. Or rather that it's dependent on the method of study used. And so now that we've outlined the hypotheses, I'm actually going to move these off to the side, because I want to take a second to actually add the totals of both columns and all the rows of the table itself. Right, so here, starting off with the number of people that passed the exam, I would add 28, 35, and 40. Which gives me a total of 103 people that passed. So to see the number of people that failed the exam, I would add 1215, and 5. Which gives me 32. And so now going across for each row would give me the total number of people that used each study method. For self-study, for example, that's 28 added to 12, which gives me 40. For the online course that's 35 added to 15. Which gives me, let's see 50. And then for the classroom course, that's 40 added to 5, which gives me 45 in total. And so the total number of people. That were included in this study. Comes out to 135. Because now it's this information that we're going to need to calculate the expected frequencies. So recall that the expected frequency or capital E as I'm writing here, is going to be equal to, in this case, the road total. Multiplied by the column total. And then divided by the grand total. So the number of people included in the survey here. So now let's begin with. Those individuals that decided to self-study. So for each method of study, we can calculate the expected frequency of those that passed and those that failed, right? So for self study. The people that passed, that is. That would be the row total of 40. Multiplied by the column total. Of 103, because again that's 103 people that passed divided by. 135, which is a grand total. And this comes out to 30.52. And now we do the same thing for the number of people that failed using the self-study method. So that would be the column total of 40 still, but multiplied by 32 people that failed the exam, and divided by the same grand total of 135. This gives you 9.48 as the expected frequency. So now let's continue with the number of people that took the online course. For the people that took the online course, the expected frequency of those. For those that passed would be. It's row total, so that would be 50. Added to excuse me, multiplied by 103. And then divided by 135, which gives you 38.15. For fail that would be 50. Multiplied by 32. Divided by 135. Which equals 11.85? And last but not least, we have the people that took a classroom course. For those that passed The expected frequency would be. 45 multiplied by 103 divided by 135. Which equals 34.33. And for fail, that would be. 45 multiplied by 32. Divided by 135, which equals 10.67. All right, so now let's take a step back to recall what we just calculated and what needs to be calculated now. Right, so at this point, for each method of study and for each outcome of the exam, we have now calculated the expected frequency. So, when looking at the table provided, you're already given the observed frequency, because you already know how many people either passed or failed using each study method. At this point now We have to calculate the chi squared value. And so recall that the chi squared value. Is equal to The sum of. The difference between the observed and expected outcome squared, divided by the expected outcome. Now the reason why I say the sum of is because the observed outcome. For each. Method, right, and for each outcome itself. Is compared to the expected outcome that was just calculated. And then the difference between them is squared before it's divided by the expected outcome. Or frequency, frequency is a correct word there, not sure why I keep saying outcome. But regardless, right, after doing so. For each data point, you would then add them together to achieve the chi squared value. Once you do so, right, once you add up the differences between all. Observed an expected frequency squared divided by the expected frequency. And then add them all together. That total would be 5.92. That is your chi squared value. So now it's 5.92 that you would then compare to your critical value. And the critical value you can use or you can find with the chi square table. So first, let's find degrees of freedom. Right, the degrees of freedom in this case. Would it be equal to the number of rows subtracted by one multiplied by the number of columns. Subtracted by one. So this would be 3 subtracted by 1, multiplied by 2 subtracted by 1, which just gives you 2. And the significance level or value, or excuse me, alpha, as mentioned before, is 0.05. So at alpha equals 0.05, and with two degrees of freedom. Chi squared critical is 5.991. So here The calculated value, 5.92. Is less than Our critical value of 5.991. Which means ultimately that we fail to reject the null hypothesis. So now this brings us finally to our answer, which reads as follows. We failed to reject the null hypothesis. There is not enough evidence to suggest study method affects exam results. Structured courses do not show a significant advantage over self-study. 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

Hypothesis Testing

Chi-Square Test of Independence

Significance Level

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