Gender and Eye Color The following table describes the distrib...

Question

Gender and Eye Color The following table describes the distribution of eye colors reported by male and female statistics students (based on data from “Does Eye Color Depend on Gender? It Might Depend on Who or How You Ask,” by Froelich and Stephenson, Journal of Statistics Education, Vol. 21, No. 2). Is there sufficient evidence to warrant rejection of the belief that gender and eye color are independent traits? Use a 0.01 significance level.

Accepted Answer

A group of students was asked about their favorite juice flavor. The results were categorized based on whether the student was in the morning class or the evening class, and were given a table with the morning and evening class and the different juice flavors in their accounts. At a 0.01 significance level, test the claim that juice flavor preference independent of class time. We have 4 possible answers, where the data do not allow a comparison, the test is inconclusive, juice preference is not independent, or there is insufficient evidence to conclude that juice preference depends on class time. Now, we're going to try to use a chi square test for independence. So we'll have to check our expected value counts, but let's first note our hypotheses. We have our no hypothesis. Where Choose preference. In class time Are independent And the alternative that they are not independent. Now let's find our row and column totals first. For the apple, we have 10 + 5, which gives us 15. Grape, 6 + 9, which gives us 15, and orange, 4 + 6 gives us 10. We can also add the row totals. 10 + 6 + 4 for the morning gives us 20. And 5 + 9 + 6 for the evening also gives us 20. Our overall total is then 40. Now let's make our expected value table. Our expected count will be equals to the row total. Multiplied by the column total. Divided by the grand total. So for example, For morning class with apple juice, a row total. It is 2 Our column total is 15, we divide this by 40. We would end up getting a value of 7.5 for morning. Apple juice. Now we'll repeat this for all of our values, and I'll go ahead and make our new expected value table. If apple We have in the middle. Grape and the last orange. And we have morning and evening. We already got 7.5 for Morning Apple, and we can just repeat this process for all the rest. 7.5. Morning orange is 5.0. Evening apple 7.5. Evening grapes, 7.5. An evening orange 5.0. So, because all of our expected counts are greater than equals a 5, we can use this test. Let's go ahead and find chi square statistic now. I be the sum. Of our observed value minus our expected value squared divided by our expected value. And so our reserve value for our first morning apple square. Will be 10 minus 7.5 squared divided by 7.5. We repeat this for all of them. We have 6 minus 7.5 squared, divided by 7.5. Plus Our morning orange, which is 4 minus 5 squared. Divided by 5 Plus 5 minus 7.5 squared. Divided by 7.5, plus our next one. Being 9 minus 7.5 squared. Divided by 7.5. Plus 6 minus 5 squared divided by 5. We add all these together, we get a value. Of 2.667 for our statistic. Now we need our critical value. Now we need our degrees of freedom first, which is our row total, -1, multiplied by our column total, -1. In this case, we have 2 rows and 3 columns. This will give us 1 multiplied by 2, or just degrees of freedom of 2. Also, at significance level of 0.01, which means our critical value. Will be at alpha equals 0.01 with degrees of freedom of 2. From the table, this corresponds with the value of 9.210. Now, you know, our value, 2.667 is less than the critical value 9.210. This means we fail. To reject The the hypothesis. Now, because we failed to reject all hypothesis, if we go back to our answers, we can then conclude The answer to our problem is answer D. There is insufficient evidence at the 0.01 significance level to conclude that juice preference depends on class time. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Key Concepts

Chi-Square Test of Independence

Significance Level

Contingency Table

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