Chapter 26 Part 2 Comparing Counts

Mars Candy Company wrote in an email to an AP Statistics teacher on the AP Stat Listserve: Our color blends were selected by conducting consumer preference tests, which indicate the assortment of colors that pleased the greatest number of people and created the most attractive overall effect. On average, our mix of colors for M&M’s milk chocolate candies is: 24% cyan blue, 20% orange, 16% green, 14% bright yellow, 13% red, 13% brown. A teacher decided to test this claim with a random sample of M&M’s candies. The sample included 50 blue, 39 orange, 34 green, 27 yellow, 28 red, and 25 brown. Is this test consistent with the company’s stated proportions?

Step 1 : State the hypotheses Blue Orange Green Yellow Red Brown Total 50 39 34 27 28 25 203 Step 1 : State the hypotheses 𝐻 0 :𝑇ℎ𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑠 𝑐𝑙𝑎𝑖𝑚𝑒𝑑 𝑏𝑦 𝑀𝑎𝑟𝑠 𝐶𝑎𝑛𝑑𝑦 𝐶𝑜𝑚𝑝𝑎𝑛𝑦. 𝐻 𝐴 :𝑇ℎ𝑒 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑠 𝑐𝑙𝑎𝑖𝑚𝑒𝑑 𝑏𝑦 𝑀𝑎𝑟𝑠 𝐶𝑎𝑛𝑑𝑦 𝐶𝑜𝑚𝑝𝑎𝑛𝑦. Step 2 : Check Conditions and Model The data consists of counts. The sample is random or representative. M&M’s candies are independent. Expected counts: .24(203)=48.72; .2(203)=40.6; .16(203)=32.48; .14(203)=28.42; .13(203)=26.39. All expected counts are greater than 5. We will do a chi-squared test for goodness-of-fit with 6-1=5 degrees of freedom.

P-value = DISTR8: 𝑋 2 cdf (0.41, 999, 5) = 0.995 Blue Orange Green Yellow Red Brown 50 39 34 27 28 25 (48.72) (40.6) (32.48) (28.42) (26.39) (26.39) Step 3: Mechanics X 2 = (50−48.72) 2 48.72 + (39−40.6) 2 40.6 + (34−32.48) 2 32.48 + (27−28.42) 2 28.42 + (28−26.39) 2 26.39 + (25−26.39) 2 26.39 =0.41 P-value = DISTR8: 𝑋 2 cdf (0.41, 999, 5) = 0.995

Step 4: Conclusion The P-value of 0.995 is very high, so we do not reject the null hypothesis which states that the distribution is as claimed by the Mars Candy Company. We do not have evidence to suggest that the distribution is different from the claimed distribution.

Be careful about assuming causation! A small P-value is not proof of causation!

Assignment: Chapter 26 HW – page 643 #10, 14, 16, 18, 20, 24 Ch 26 Homework Due Wednesday M&M’s on Monday!