Inference – Dist. by Categories

How likely is it that differences this large or larger would occur just by chance in random samples of size 60 from the population distribution claimed by Mars?

To make the comparison, we use the chi-square statistic χ2 = ∑ (observed – expected)2

Expected

This is the measure of the distance of the observe counts from the expected counts

Large values of χ2 are strong evidence against H0 because the observed counts are far away from expected Small values of χ2 suggest that the data are consistent with the null hypothesis

Inference – Dist. by Categories

Jerome’s class did the M&M Activity and here are his results:Calculate the chi-square statistic

We divide each category by its respective expected value so that the largest relative difference contributes more heavily to the evidence against the null

Color Observed Expected

Blue 9 14.4

Orange 8 12

Green 12 9.6

Yellow 15 8.4

Red 10 7.8

Brown 6 7.8

Inference – Dist. by Categories

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The chi-square distribution includes only positive and is skewed right.

Each chi-square distribution is specified by giving its degrees of freedom. Degrees of freedom for a chi-square goodness-of-fit test = number of categories – 1

As the degrees of freedom increase, the density curves become less skewed, and larger values become more probable

The mean of a particular chi-square distribution is equal to its degrees of freedom

When degrees of freedom are > 2, the peak of the chi-square density curve is at df – 2

Inference – Dist. by Categories

To find the P-value from a chi-square distribution, we use Table C

Use the degree of freedom row on the left of the table

Locate the approximate values the χ2 lies between and the P-value lies between the two values at the top of those two columns.

Inference – Dist. by Categories

In the last example, we computed the chi-square statistic for Jerome’s random sample of 60 M&M’s Milk Chocolate Candies: χ2 = 10.180. Now let’s find the P-value. Because all the expected counts are at least 5, the χ2 statistic will follow a chi-square distribution reasonably well when H0 is true. There are 6 color categories for M&M’s Milk Chocolate Candies, so df = 6 -1 = 5.

The P-value is the probability of getting a value of χ2 as large as or larger than 10.180 when H0 is true.

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Inference – Dist. by Categories

Technology can give us a more precise P-value

DISTR – χ2 cdf χ2 cdf ( χ2 , large # (1000), df )

Since our P-value is 0.07 is greater than our significance level α = 0.05, we fail to reject H0.

We don’t have sufficient evidence to conclude that the company’s claimed color distribution is incorrect.

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