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

According to the manufacturer of M\&Ms, $13 \%$ of the plain M\&Ms in a bag should be brown, $14 \%$ yellow, $13 \%$ red, $20 \%$ orange, $24 \%$ blue, and $16 \%$ green. A student wanted to determine whether a randomly selected bag of plain M\&Ms had contents that followed this distribution. He counted the number of M\&Ms that were each color and obtained the results shown in the table. Test whether plain M\&Ms follow the distribution stated by M\&M/Mars at the $\alpha=0.05$ level of significance. CAN'T COPY THE TABLE

Key Concepts

p-value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic at least as extreme as the one observed. It is used to decide whether the observed differences are statistically significant – a p-value lower than the significance level leads to rejection of the null hypothesis.

Expected Frequency Calculation

Expected frequencies are calculated by multiplying the total sample size by the hypothesized probability for each category. These expected counts serve as the benchmark against which the observed frequencies are compared in the test.

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 crucial in determining the critical value from the chi-square distribution, which is then used to evaluate the test statistic.

Chi-Square Goodness-of-Fit Test

This statistical test is used to determine whether the observed frequencies of categorical data match the expected frequencies based on a given theoretical distribution. It involves computing a test statistic that measures the discrepancy between observed and expected counts to assess the overall fit of the model.

Alternative Hypothesis

The alternative hypothesis posits that there is a significant difference between the observed and the expected frequencies, indicating that the data do not follow the specified distribution. This hypothesis is tested against the null in determining the validity of the model.

Null Hypothesis

The null hypothesis in a goodness-of-fit test states that the observed distribution does not differ significantly from the expected distribution. Essentially, it claims that any differences between observed and expected frequencies are due to random chance.

Test Statistic

The test statistic for the chi-square goodness-of-fit test is computed by summing the squared differences between observed and expected frequencies, divided by the expected frequencies for each category. This statistic quantifies the deviation of the observed data from what is expected under the null hypothesis.

Significance Level

The significance level, denoted by alpha (?), is a threshold set by the researcher (commonly 0.05) to decide whether to reject the null hypothesis. It represents the probability of committing a Type I error, or rejecting the null hypothesis when it is actually true.

Transcript

00:01 So we're going to assume that the distribution is correct, that the mars candy.

00:09 And you could go through and tell that the proportion of browns is equal to 13%, etc., and list them all down.

00:16 And alternately, the distribution that is quoted is not correct.

00:23 So that at least one of the proportions is off.

00:26 And we need to calculate that kye squared statistic.

00:29 Now we had six categories, so we'll have five degrees of.

00:32 Freedom and i went through and added up the total number of m &ms by the way and we had the observed value of all those m &ms and those added up to 400 and then we had all of those percents that were given and so i'm finding my expected values and i'll list on what those expected values are that i had to take each of these 400 times each of those expected values and these came out to be 52, 56, 52, 80, 96, and 64.

01:14 So those were the corresponding values, and then you had all the other values that were for the different colors given.

01:19 So when we calculate that test statistic, i'm going to take that first value minus the expected value square divided by the expected.

01:28 And then my second value was 64, and i think that was for yellow, minus 56 quantity square divided by the expected, and continue to do that for all of those events.

01:41 And then i will, i have all this all in my calculator, and i'm ready to do my calculation.

01:49 And we find that, again, we have five degrees of freedom, and we find that the kye squared statistic i get is 7 .5 .5.

01:59 7 .29...

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