Using the Kruskal-Wallis Test
In Exercises 5–8, use the ...

Question

Using the Kruskal-Wallis Test

In Exercises 5–8, use the Kruskal-Wallis test.

HIC Measurements Use the sample data from Exercise 1 with a 0.05 significance level to test the claim that small, midsize, large, and SUV vehicles have the same median HIC measurement in car crash tests.

Accepted Answer

All right, hello, everyone. So this question says, a researcher collected data on the drying times in minutes of 4 types of wall paint matte, satin, semigloss, and gloss. The data were collected under controlled temperature and humidity conditions. The drying times in minutes are listed below. At the 0.05 significance level, use the Cresco Wallace test to determine whether there is a significant difference in the median drying times among the four paint types. All right, so first and foremost, let's begin with stating our hypotheses, right? Starting off with the null hypothesis. The null hypothesis would state that all four paint times, or excuse me, all 4 paint types have the same drying time. Or at least the same median drying time. So therefore, the alternative hypothesis would state the opposite, right, which means that at least one pain type. Is different in terms of it having a different median drying time. So here on the screen, I went ahead and wrote down the. 20 values in a table format so that we can go ahead and assign ranks to them in average ranks in the case of ties. Now, recall that a tie just depends on whether or not the same value is repeated in a data set. So, 38, for example, because it's only seen once in the data set and it's the smallest number, it's going to have an average rank of 1, and it's also going to have an average rank of 1, because there are no repeats. 39 for the same reason would have the rank of 2. But 40 on the other hand. Twice Is there a drying time of 40 minutes recorded, which means that the number 40 itself takes places 3 and 4, which means that the average rank would be the average of 3 and 4, which gives you 3.5, right, because that's 3 added to 4 and divided by 2. So now for 41, that would be ranks 5 and 6, because again it's repeated twice, with an average rank of 5.5. 42 would have a rank of 7 because it's not repeated, so the average rank is also 7. Now 43 is actually repeated twice, so it would take places 89, and 10. And so the average rank of 43 would be the sum of 89, and 10 divided by 3, which gives you 9. So for 44, that places 11 and 12, or rather ranks 11 and 12, giving you an average of 11.5. 45 would just be rank 13. And from this point onwards, all of the data points are repeated only once. So from 46 to 52, we have ranks 14. To 20 respectively. And so the average rank would just be the same thing. So now it's at this point that you're going to sum up the rankings by group. So for each paint type, right, you're going to. Lists, rather, the average ranks of each of the data points that correspond to that pain type. So for matte paint, for example, the corresponding average ranks would be 12. 3.5, 5.5, and 9. And then in the next column of this table that I drew here, you're going to add together all of the average ranks. So for matte paint that gives you 21. Similarly, for satin paint, the ranks are 11.5. 15 1314 and 9. And the sum here would be 62.5. Semigloss Corresponds to the ranks of 18. 1719, 16 and 20. And so the sum here would be 90. Last but not least, you have gloss, which is 7 3.5, 5.5. 9 and 11.5. And this gives you a total of 36.5. So from here, we move on to the Cresco Wallace formula, which is as follows, right? H is equal to 12 divided by N, multiplied by N added to 1. Multiplied by the sum of R1 square divided by N1 and subtracted by 3 multiplied by. Capital N rather added to 1. Now the lower case n here. In the denominator of our summation is going to be equal to 5 for all groups. is actually equal to the number of ranks in each group, so that's equal to 5 for all of them. Capital N on the other hand. Is going to be equal to 2 now are. Our sub I corresponds to the group rank sums that we determined earlier. So plugging in the information that we are given, and the information that we calculated, H is equal to the following. Right, that's 12 Added to 20 multiplied by 21. And then multiplied by. The sum of. 21 squared over 5 62.5 squared over 5. 90 square over 5. And 36.5 squared over 5. This is also multiplied by 3, multiplied by 21. So H therefore, is approximately equal to 15.74. Which we now compared to the critical value. Now for this question, right, the. Degrees of freedom Is equal to 3. And as mentioned in the text of the question, alpha, that's the confidence level, is 0.05. And so the critical chi square value with this information in mind is 7.815. So, just to put everything back into perspective, because our calculated value is greater than the critical value, we reject the null hypothesis. So now this brings us to our final answer, which reads as follows. There is sufficient evidence to conclude that at least one paint type has a different median drying time. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

Using the Kruskal-Wallis Test
In Exercises 5–8, use the Kruskal-Wallis test.

HIC Measurements Use the sample data from Exercise 1 with a 0.05 significance level to test the claim that small, midsize, large, and SUV vehicles have the same median HIC measurement in car crash tests.

Key Concepts

Kruskal-Wallis Test

Median

Significance Level

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