Stat 100, Mar. 13 Read Chapter 18, Try problems 1-6, 10-11

A preliminary- During this lecture we’ll use this result from earlier in the term: For normal curve (bell-shaped): About 68% is in interval mean  SD About 95% is in interval mean  2 SD About 99.7% is in interval mean  3 SD

Imagine coin flipping activity Everybody in the class flips a coin 64 times Each person finds proportion of his/her flips that are Heads We draw a histogram of the results.

Coin flip example continued Are some possibilities for the proportion (heads) more likely than others? What would be the shape of the histogram of the proportions from the different class members?

Rule For Sample Proportions Normal curve approximately describes possible sample proportions that could result from the many different possible samples in a random circumstance.

Characteristics of this normal curve Mean = “population” or theoretical value of the proportion (p) Std Dev =

Consider n=64 coin flips Let p=proportion that are heads True p = 0.5 Mean of curve for possible sample proportions is 0.5 St Dev is sqrt (0.5 x 0.5/64) = 0.06

From % Rule About 68% chance that sample proportion will be in range 0.5±0.06, or 44% to 56% About 95% chance that sample proportion will be in range 0.5±0.12, or 38% to 62% About 99.7% chance that sample proportion will be in range 0.5±0.18, or 32% to 68%

Application Suppose n=64 people each tasted Pepsi and Coke in a “blind” experiment. The result is that 40/64=0.625, or 62.5% state a preference say the Coke tasted better.

Pepsi/Coke example Suppose everybody really picks preferred drink randomly So, picking preferred drink is like a coin flip. For instance, picking Coke might be like flipping Heads Results from n = 64 people would be like results of 64 coin flips

Pepsi/Coke continued For 64 coin flips, we found that about 95% of the time the proportion “heads” will be in interval 38% to 62% Proportion will be greater than 62% only about 2.5% of the time So, 62.5% picking Coke would be unusual if preferences are random (like coin flips) Inference: Data might be evidence that preferences aren’t random

A past class activity Students picked a number between 1 and 10. N=123 participated 29 students picked the number 7. Sample proportion is 29/123 =.24 (24%)

For true random picks Proportion picking “7” would be p=1/10 =.1 Mean sample p =.1 Std. Dev = sqrt (.1 x.9 / 123) = 0.027

Do Students Pick Randomly? With random picks, chance is 99.7% that proportion picking “7” will be in interval.1± (3 .027), or.019 to.181 The class result =.24 picking “7” is well outside this interval. Inference: Students probably don’t pick randomly - observed result very unlikely to result from random selection