Arsenic in Rice Listed below are amounts of arsenic in samples...

Question

Arsenic in Rice Listed below are amounts of arsenic in samples of brown rice from three different states. The amounts are in micrograms of arsenic and all samples have the same serving size. The data are from the Food and Drug Administration. Use a 0.01 significance level to test the claim that the three samples are from populations with the same median.

Accepted Answer

All right. Hello, everyone. So this question says, a quality control officer measured the hardness in Brunel units of steel bolts manufactured at 3 different factories, factory X, factory Y, and factory Z. Each bolt was tested under the same conditions, and the measurements are listed below. At the 0.01 significance level, use the Cresco Wallace test to test the claim that the three factories produce bolts with the same median hardness. All right, so first, what are the hypotheses? Right? First we have the no hypothesis. Which would state that all three factories produced bolts with the same median hardness. The alternative hypothesis, on the other hand, would state that at least one factory has a different median hardness. So here I went ahead and made a table with all of the values listed in the data set. Because what we're going to do here is rank them in order from smallest to largest, and then calculate the average rank as necessary. Now, what I mean by that, for example, is starting off with the first value of 85. 85 happens to be or happens to appear twice in the data set. And so because 85 is the smallest possible value, and it appears twice, it would take the ranks of both 1 and 2. Which means that the average rank is the average of 1 and 2, so that's 1 added to 2, and divided by 2, which gives you 1.5. The same is true for 86, right? 86 happens to appear twice, so it would take ranks 3 and 4, with an average rank of 3.5. 87 would have 5 and 6, so that's an average of 5.5. 88 takes 7 and 8. So that's an average rank of 7.5. And 89 takes 9 and 10. So that's 9.5. The next set of values are from 91 to 95, and all of these happen to appear only once, which means that their average rank is simply equal to their ordered rank. So that's Ranks 11 through 15. Which equals the average rank for each value. So now, our next step is to find the sum of ranks for each factory, right? So what we're going to do here is list the ranks of the data points that correspond to each factory, and then add the sum of all the ranks together, which corresponds to this R. Or of I value So for factory X, the ranks are 1.5. 3.5 5.5. 7.5 and 9.5, which gives you a total of 27.5. For factory Y that's 11. 1213, 14, and 15. Which gives you 65 in total. And then for factory Z, that's 1.5, 3.5. 5.5, 7.5, and 9.5, giving you a total of 27.5. So now The next step is to apply the Cresco Wallace test statistic formula. Which gives you H is equal to 12 divided by N multiplied by N added to 1, multiplied by the sum of. The rank sums here for each category, squared multiplied by lowercase n sub 1, and then subtracted by 3 multiplied by capital N added to 1. So, in this context, lowercase n corresponds to the number of rankings per category, which would be 5 for each group here, and uppercase N is equal to 15. So here Age is equal to 12. Multiplied by 15. Or divided by excuse me 12 divided by 15 multiplied by 16. And then multiplied by. 27.5 square divided by 5. Added to 65 square divided by 5. And added to 27.5 square divided by 5. This is all subtracted by 3, multiplied by 16. So after evaluating this expression, capital H is equal to approximately 9.37. Which you would then compare to the critical value. So here The degrees of freedom would be equal to K subtracted by 1. K is equal to 3. So that's 3 subtracted by 1, which equals 2. And as mentioned before, alpha, which is the confidence interval, is equal to 0.01. So from the chi square table, at 2 degrees of freedom and alpha equal to 0.01, the critical value. Is equal to 9. 9.210. So now, the last step is to compare. And because our calculated value of 9.37 is greater than the critical value of 9.210, we can reject the null hypothesis. So now this brings us to the final answer, which reads as follows. There is sufficient evidence to conclude that at least one factory produces bolts with a different median hardness. 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

Median

Hypothesis Testing

Significance Level

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