According to the manufacturer of M&Ms, 13 % of the plain M & Ms in a bag should be brown, 14 % y

step 1 Okay, so let us look at this question. Now, this question says that according to the manufacture

According to the manufacturer of M&Ms, 13 % of the plain M &...

Key Concepts

Degrees of Freedom

Degrees of freedom in a chi-square goodness-of-fit test are calculated as the number of categories minus one. This value is used to determine the critical value of the chi-square statistic from the chi-square distribution for a given significance level.

Expected Frequency

Expected frequencies are calculated based on the total number of observations and the hypothesized proportions for each category. These values represent the number of occurrences that would be expected in each category if the null hypothesis were true.

Significance Level

The significance level, often denoted by ?, is the threshold for deciding whether to reject the null hypothesis. It represents the probability of making a Type I error. In this context, a significance level of 0.05 indicates a 5% risk of rejecting the null hypothesis when it is actually true.

Chi-square Goodness-of-Fit Test

This test is used to determine whether observed frequencies in categorical data differ significantly from expected frequencies derived from some theoretical or hypothesized distribution. It helps in assessing the fit of the observed data to the expected proportions.

Hypothesis Testing

Hypothesis testing involves formulating null and alternative hypotheses. In the context of the chi-square goodness-of-fit test, the null hypothesis typically states that the observed data follow the expected distribution, while the alternative hypothesis states that there is a deviation from this distribution.

Transcript

00:01 Okay, so let us look at this question.

00:03 Now, this question says that according to the manufacturer of m &ms, 13 % of the plain m &ms in the back should be brown.

00:12 And then you have different percentages for yellow, red, blue, orange, and green.

00:17 Okay, so let us just look at the table that we have.

00:21 The first and the foremost thing we should do while solving a kai square problem is draw the table.

00:29 So we have colors and then we have frequency.

00:33 We observe frequency.

00:36 This is frequency.

00:43 All right.

00:46 So the different colors that we have are brown, yellow, red, blue, orange, green.

00:50 So this is brown, yellow, red, blue, orange, and green.

01:13 All right.

01:15 Now, what are the observed frequencies that we have? 61 64 54 61 64 64 61 96 64 64 61 96 64 61 96 64 61 96 and then we have 64 okay so a student randomly selected a bag of plain m &ms he got at the number of mnms right and this is the frequency that we have the frequency table now what we have to do is we have to test whether the plain m &ms follow the distribution stated by the mnm mars.

02:05 Okay, 13 % of plain mnms in a bag should be brown.

02:11 All right.

02:12 Okay, so our alpha is 0 .05.

02:15 This is a simple question.

02:18 So just a moment.

02:22 Okay, so alpha, okay, what is happening? yeah.

02:36 So this for some reason is not working, but okay, we'll make it work.

02:40 So alpha, what is this not working? just a moment, guys, just bear with me.

03:08 This should not happen.

03:18 Okay, yeah, it's good.

03:20 So alpha is 0 .05.

03:24 Now, what is a null hypothesis? the null hypothesis, h0, is going to be that the plane is for the distribution stated by the mnmars.

03:37 Okay, that the plane...

03:42 M &ms follow the distribution given by the manufacturer m &m mars given by mnm.

04:08 How do you write this? m &m mars.

04:11 Okay, i think this is the manufacturer.

04:14 What is going to be the alternative hypothesis? the alternative hypothesis will be that the plain mnms, m &m's, don't follow the distribution given by mnm mars.

04:42 All right.

04:44 Now, what we have with us are different categories and the observed frequencies.

04:50 So from the observed frequencies, what we can do is we can find the sample size, right? what is the sample size? it is going to be the addition of all of these frequencies.

04:59 So let me do one thing.

05:01 Let me take the calculator.

05:02 Let me add all of these.

05:03 61 plus 64 plus 54 plus 61 plus 96 plus 64 which is 400 so this is 400 okay now what are the probabilities what are the probabilities right what are the probabilities the manufacturer says that 13 % should be brown right so this should be 13 % or let me write this as 0 .13.

05:35 14 % should be yellow, 13 % should be red.

05:38 So, yellow is 14 .14.

05:41 Red is 0 .13.

05:45 24 blue, 20 orange.

05:47 So this is 24 blue, 20 orange.

05:52 And 16 should be green.

05:53 So this is 0 .16.

05:58 Now, in order to conduct the kai square test, we need to find what is known as the expected values.

06:07 Right, we need to find what are the expected values if the mnms actually follow this distribution, this distribution in this column.

06:17 So how do you find that? expected values, let us come down here.

06:22 Expected values for any category are given by the sample size, which is 400 in our case, the sample size, multiplied by the probability for each category, multiplied by the probability, probability.

06:39 Probability for each category or for category i in this case right because we are finding e i expected value for the iath category so let us apply this formula here now i have 400 so what is the probability for brown it is 0 .13 so what should be the expected value 13 % of 400 should be brown okay so 13 % of 400 is what this is 50 if i'm not wrong yes so this is 52 is similarly this will also become 52 now this is 14 right so 14 percent of 400 is 56 okay point 2 point 2 is for 400 right this should be 80 point 24 what will be point 24 so this is going to be 24 into 4 which is nothing but 96 96, okay, and 0 .16.

07:52 So this is 4 into 16, which happens to be 64, right? yeah, this is 64.

08:05 All right.

08:05 Now that we have the expected values.

08:08 Now we have to calculate the kai square statistic and what is a formula for that? to calculate the kai square statistic, for every category, you do this computation, you find the difference between the observed and the expected value, you square the difference, divide the difference by the expected value, and after doing this for all the categories, you simply sum them all up.

08:30 So this is what you are going to do here.

08:32 Let us look at this formula in action.

08:36 So the first category is brown, right? so what is the difference between the observed frequency and the expected frequency? 61 minus 52...

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