Question 12 In a large college, it is reckoned the numbers...

Question 12 In a large college, it is reckoned the numbers of students with grey, blue, brown or green eyes are in the ratio 8: 4:5:3. 160 students are selected at random and the numbers with each eye colour are shown in the table below. Eye colour grey blue brown green Frequency 71 23 48 18 [1] (a) State your hypotheses. (b) Work out the expected frequencies. [1] (c) Carry out a chi-squared test at a 5% level of significance. What conclusion can you draw? Would you conclusion be different if the [6] level of significance is 10%? Explain your answer.

Transcript

00:01 For this problem to begin, i'll note that i can see we kind of have a wall of text here, but basically we're doing a chi -squared test.

00:07 So what i'll do is i'll just go through a thorough sort of run -through of a chi -squared test for independence, or it can also be approached as chi -squared test for goodness of fit.

00:17 But i'll just go through all of the steps, and we should be covering all of the different details that you are asked about in each one of the individual parts of the problem there.

00:26 So the null hypothesis is that the distribution of eye color is uniform.

00:33 So we can say that the null hypothesis is a uniform distribution.

00:44 So we'd say that proportion in category one, which we'll say is blue, is equal to the proportion in category two, is equal to the proportion in category three, is equal to the proportion in category four.

00:55 Since we have four categories, and we're assuming that all are equally likely, or there's an equal amount in each category, we'd say that the expected proportion for each category is just one over four.

01:06 The alternative hypothesis would be that the data is from a not uniform distribution.

01:17 To determine our, or pardon me, let me rephrase this.

01:21 So the test that we'll be doing here is, as i said, it's kind of chi -squared test for independence, kind of chi -squared test for goodness of fit, just depending on how you look at it.

01:30 But ultimately, the process here is going to be the same.

01:33 Our chi -squared statistic will be the sum from i equals 1 to 4 of the observed frequency for category i minus the expected frequency for category i squared divided by the expected frequency for category i.

01:47 We have that the expected frequency for each category is just going to be the total number of students in the sample divided by 4 since we expect that there's an even proportion in each.

01:57 So i'll be using a ti -84 plus here for my calculations, including i will be using the built -in chi -squared test functionality, but i'll sort of show one part of the calculation first because it's a sort of tedious process here.

02:11 So first we want to find the total number of observations.

02:16 We want the total number of observations, which we'd find by taking, let's see here, we have 3 blue plus 2 green plus 12 brown plus 3 hazel.

02:26 So we have 20.

02:27 20 over 4 would be 5.

02:31 So we would expect to have exactly 5 individuals in each one of the categories.

02:36 So, for instance, for that first term in our chi -squared, in the calculation of the chi -squared statistic, we know that that is going to be observed frequency 1 minus expected frequency 1 squared divided by expected frequency 1.

02:51 We do plus dot dot.

02:54 That first observed frequency was three.

02:57 First expected frequency is five...

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