4.2 – P-VALUE - Hypothesis Testing - Procedure and Reasoning

1)There is a claim that a coin is favoring heads. (Ha: p > 0.5).

2)Let’s analyze the case of a fair coin:

If the coin is fair we expect getting heads 50% of the time. (Ho: p = 0.5). A simulation for 1000 repetitions of “flip a coin 10 times and record the number of heads obtained” is shown below.

/ Find the p-value if we flip the coin 10 times and obtain
a)10 heads
b)9 heads
c)7 heads

3)Based on these probabilities we think: Does this look unusual for a fair coin? Is this result more likely to happen when the coin is favoring heads? Is this convincing evidence that the coin favors heads?

What do we conclude? If we obtain 10 heads in 10 flips; which one is the correct conclusion?

(1)It’s not unusual to obtain 10 heads in 10 flips of a fair coin; we don’t have convincing evidence that the coin favors heads

(2)It’s unusual to obtain 10 heads in 10 flips of a fair coin; it’s more likely for this to happen when the coin favors heads; so we conclude that there is evidence to say that the coin favors heads.

What do we conclude? If we obtain 9 heads in 10 flips; which one is the correct conclusion?

(1)It’s not unusual to obtain 9 heads in 10 flips of a fair coin; we don’t have convincing evidence that the coin favors heads

(2)It’s unusual to obtain 9 heads in 10 flips of a fair coin; it’s more likely for this to happen when the coin favors heads; so we conclude that there is evidence to say that the coin favors heads.

What do we conclude? If we obtain 7 heads in 10 flips; which one is the correct conclusion?

(1)It’s not unusual to obtain 7 heads in 10 flips of a fair coin; we don’t have convincing evidence that the coin favors heads

(2)It’s unusual to obtain 7 heads in 10 flips of a fair coin; it’s more likely for this to happen when the coin favors heads; so we conclude that there is evidence to say that the coin favors heads.

Section 4.2: Measuring Evidence with P-values

1)Go to the book, section 4.2 (p. 236) or the power point presentation for section 4.2, and complete the following:

What is the p-value?

It’s the probability, when the ______is true, of obtaining a

______as extreme, or more extreme than ______

2)Which P-value shows more evidence against Ho and in favor of Ha?

In each case, which p-value provides the strongest evidence against H0 and for Ha?

a). p-value = 0.95 or p-value = 0.02

b). p-value = 0.008 or p-value = 0.02

3)Complete the following:

The smaller the p-value the ______the statistical evidence against ______

and in favor of ______

4)Do dogs resemble their owners?First read the beginning of section 4.1 (page 220) through example 4.1. Continue reading through the end of page 222 (stop at Smiles and Liniency)

Dog and Owners again: Continue reading examples 4.13, 4.14 in section 4.2 (page 237)

a)Write the two hypothesis

b)Look at the histograms in figures 4.7 and 4.9. Give the p-values for

  1. 16 matches
  1. 15 matches
  1. 19 matches

c)Which of the p-value provides the strongest evidence against H0 and for Ha?

d)In the study, there were 16 matches out of 25. What is the conclusion in context?

5)Support for the Death Penalty

In 1980 and again in 2010, aGallup poll asked a random sample of 1000 US citizens “Are you in favor of the death penalty for a person convicted of murder?” In 1980, the proportion saying yes was 0.66. In 2010, it was 0.64. Does this data provide evidence that the proportion of US citizens favoring the death penalty was higher in 1980 than it was in 2010? Use p1 for the proportion in 1980 and p2 for the proportion in 2010.

First let’s reflect about the problem

a) Who/what are the cases?b) One or two groups?C) Variable? d) Categorical or quantitative?

e) Objective of the study (circle in the statement of the problem)f) What is this problem about and what notation will be used? (1) One single mean? (2) One single proportion?

(3) Difference of means?(4) Difference of proportions?

a)State the null and alternative hypotheses:

b)Give the value of the sample statistic phat1 – phat 2 and locate in the graph below.

c) A randomization distribution assuming the null hypothesis is true is shown. Which of the following is closest to the p-value?

0.001, 0.05, 0.20, 0.5

d) Circle the correct statement:

(1) Results are statistically significant

(2) Results are not statistically significant

e) TRUE or FALSE? We have strong evidence against the null hypothesis and in favor of the alternative.

f) Write the conclusion in context.

g) What is the p-value for a sample statistic of 0.04? Will the conclusion be the same as in part (f)?

6)Sleep vs Caffeine for Memory

In an experiment, students were given words to memorize, then were randomly assigned to either take a 90 minute nap or take a caffeine pill. A couple hours later, they were tested on their recall ability. We wish to test to see if the sample provides evidence that there is a difference in the mean number of words people can recall depending on whether they take a nap or have some caffeine.

First let’s reflect about the problem

a) Who/what are the cases?b) One or two groups?C) Variable? d) Categorical or quantitative?

e) Objective of the study (circle in the statement of the problem)f) What is this problem about and what notation will be used? (1) One single mean? (2) One single proportion?

(3) Difference of means?(4) Difference of proportions?

a)Write the hypotheses:

b)The observed sample statistic is . Use the randomization distribution to state the p-value.

c)Under the rule: “If p < 0.05 we have statistically significant results”. Are the results of this test statistically significant? Write the conclusion in context.

d)My perception is that “a nap helps with word recall”. Write the hypotheses in this case, state the p-value and write the conclusion in context.

1