Problem

Where do women most often buy casual clothing? Da…

Problem 14 Medium Difficulty

M\&MMARS, makers of M\&M Chocolate Candies, conducted a national poll in which
consumers indicated their preference for colors. In the brochure "Colors," made available
by M\&M/MARS Consumer Affairs, the traditional distribution of colors for the plain candies is as follows:
$$ \begin{array}{lll}{\text { Brown }} & {\text { Yellow }} & {\text { Red }} & {\text { Orange }} & {\text { Green }} & {\text { Blue }} \\ {30 \%} & {20 \%} & {20 \%} & {10 \%} & {10 \%} & {10 \%}\end{array} $$
In a follow-up study, -1-pound bags were used to determine whether the reported percent-
ages were valid. The following results were obtained for a sample of 506 plain candies.
$\begin{array}{cccc}{\text { Brown }} & {\text { Yellow }} & {\text { Red }} & {\text { Grange }} & {\text { Green }} & {\text { Blue }} \\ {177} & {135} & {79} & {41} & {36} & {38} \\ \hline\end{array}$
Use $a=.05$ to determine whether these data support the percentages reported by the
company.

Answer

There is sufficient evidence to reject the claim of the specific distribution.

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Intro Stats / AP Statistics

Essentials of Modern Business Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

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Video Transcript

So in this problem, we want to test to see if the reported percentages by the company are actually true. Based on our data, so are no. Hypothesis is going to be that, um each of our proportions are equal to those reported by the company. So the proper proportion of population that's brown would be 0.3 portion of the population that's yellow would equal 0.2 proportion of the Eminem's population that is equal, that is red is equal to 0.2 proportion, That is, orange is equal to 0.1 proportion, that is green is equal to 0.1 in the proportion, that is, and blue is equal to 0.1. And now our alternative, I thought this is would be that, um, the percentages percentages reported by the company are not all the same. So we're given the following information in the problem. I just wrote this into this table here and now we have to find out the rest of our data. So the first thing I'm going to find is the expected frequency. The expected frequency is the probability of proportion given by the company times our end and our end is going to be equal to the sum of our observed frequencies and were also given in the problem that are in his 506. So in our expected frequency for the number of Eminem's that are brown would be 0.3 times 506 which is equal to um 1 51 0.8 1 51.8 and then it would be one A 1.2 101.2 50.6 and the restaurant just 50.6 because they all have the same proportions. And now the difference between the two would be or the difference would be the difference between the observed frequency and the expected frequency. So that would be 1 77 minus 1 51.8 So we get the following differences negative 12.6. And now we have Thio Square. This data and the reason we're doing this is because we are looking to find a chi squared goodness of fit and are to conduct a hypothesis test and the formula for our chi square goodness if it is the sum of, um, each of the square differences between between the, um given the actual frequency minus the expected frequency squared. Sir, I just want to get the indices the same. So it would be the expected frequency minus for the actual frequency. Modesty expected Frequency square over the expected frequency. And we're going to take some of that for all of our categorical groups, so we need to find the differences first. So, um, our first difference squared would be 6 35.4 Yeah, for 92.84 92.16 to 13.16 and 1 58.16 And now we're just going to find the value of what we're going to some for Kai squared. And that would be on. I'm going to write the truncated, um, values approximately 4.18 approximately 11 point to nine. Approximately 4.87 Approximately 1.82 Approximately 4.21 in approximately three 0.14 And now, in order to find our chi squared, we're going to take the sum of all of these values, and we get it. Kai squared of 29.5 one. And now with that kai squared of 29.51 We need to figure out our degrees of freedom so that we can come up with our P value. So our degrees of freedom, simply the number of categories that we have minus one. And in this situation we have six categories, so we have six categories brown, yellow, red, orange, green and blue. So we have five years of freedom. So our P value given that chi square is equal to 29.51 we have five degrees of freedom is equal to, um, less than point less than 0.5 So compared to an Alfa of 0.5 and the P value less than 0.5 our P value is less than 0.5 which is less than 0.5 are Alfa. So therefore, we reject the no hypothesis and what does that mean? That means that there is sufficient evidence to reject the claim that the companies percentages are correct

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