Chapter 8-A

Chapter 8: Hypothesis Testing for Population Proportions

Learning Objectives

• Formulate a statistical hypothesis concerning a population proportion.

• State the conditions required for the collection of evidence to perform a statistical hypothesis test involving proportions.

• Evaluate the evidence through a p-value calculation and interpretation.

• Decide on the statistical hypothesis based on the level of significance.

Section 8.1: The Requirements of a Hypothesis Test

Hypothesis Testing: Introduction

Hypothesis testing is a statistical inference procedure that enables us to choose between two hypotheses when we do not know all the information about the population. In hypothesis testing, hypotheses are always statements about population parameters (never about statistics).

General Steps in Hypothesis Testing

1. Hypothesize: State a hypothesis (claim), H0, that will be weighed against an alternative claim, HA.

2. Prepare: Determine what data you need to collect to make your decision and minimize the probability of making mistakes.

3. Compute to Compare: Collect data and compare them (or what they suggest) with what you would expect if the status-quo situations were true.

4. Interpret and Conclude: State your conclusion. Do you believe the data, or do you find that the claim does not have enough evidence to back it up?

Example: Coin Toss

• Suppose you hypothesize that a coin is more likely to come up heads because one side is slightly heavier.

• You decide to collect evidence by tossing the coin 20 times and counting the number of heads.

• If you get 19 heads out of 20 tosses, you would compare this result to what you would expect if the coin were fair.

• If the probability of getting 19 heads in 20 tosses is very low under the assumption of fairness, you may conclude the coin is biased.

Formulating Hypotheses

• Null Hypothesis (H0): The 'status quo' statement about a population parameter (e.g., 'no change,' 'no effect').

• Alternative Hypothesis (HA): The statement that contradicts the null hypothesis (e.g., 'there is a change,' 'the effect exists').

• Hypotheses are always about parameters, not statistics.

Example: Stating Hypotheses

• In 2000, 54% of undergraduates finished their studies with debt. A student union president believes the incidence of debt is now higher.

• H0: p = 0.54 (The proportion is the same as in 2000.)

• HA: p > 0.54 (The proportion is higher than in 2000.)

Potential Mistakes: The Significance Level

• The significance level (α) is the probability of making the mistake of rejecting the null hypothesis when it is actually true (Type I Error).

• Type I Error: Rejecting H0 when it is TRUE.

• Type II Error: Not rejecting H0 when it is FALSE.

Decision	H0 TRUE	H0 FALSE
Reject H0	Type I Error	Correct
Do Not Reject H0	Correct	Type II Error

Example: Interpreting the Significance Level

• H0: p = 0.54

• HA: p > 0.54

• The significance level is the probability of rejecting H0 when it is true.

• It is the probability of concluding that the proportion of undergraduates with debt is higher than in 2000 when, in fact, the proportion has not changed.

Section 8.2: Finding p-Values

Gathering Evidence: The Test Statistic

• The test statistic compares the observed outcome from the sample with the outcome we would get if the null hypothesis were true.

• If the test statistic is far from the value we would expect if the null hypothesis is true, we reject the null hypothesis.

Test Statistic for One Population Proportion

The test statistic has the structure:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• \( \hat{p} \): sample proportion

• \( p_0 \): proportion stated in the null hypothesis

• \( n \): sample size

Example: Computing the Test Statistic

• Suppose 585 out of 1000 recent graduates have debt. Test H0: p = 0.54 vs. HA: p > 0.54.

• \( \hat{p} = 0.585 \), \( p_0 = 0.54 \), \( n = 1000 \)

• $z = \frac{0.585 - 0.54}{\sqrt{\frac{0.54 \times 0.46}{1000}}} = 2.358$

• The observed value of the test statistic is 2.358. This means the observed proportion is 2.358 standard deviations higher than expected if the null hypothesis were true.

Measuring Surprise: The p-Value

• The p-value is the probability, assuming the null hypothesis is true, of obtaining a result as extreme or more extreme than the one observed.

• A small p-value (typically less than α) suggests that a surprising outcome has occurred, and we reject the null hypothesis.

p-Values and the Significance Level

• If p-value < α: Reject H0

• If p-value ≥ α: Fail to reject H0

Example: Interpreting the p-Value

• Suppose the p-value is 0.0091 for the test statistic z = 2.358.

• This means that, if the null hypothesis is true, the probability of getting a sample proportion as high as 0.585 or higher is 0.91%.

• This is strong evidence against the null hypothesis.

Conditions for Using the Test Statistic

• Random sample

• Large sample size: n × p0 ≥ 10 and n × (1 - p0) ≥ 10

• Large population: Population is at least 10 times larger than the sample size

• Independence: Each observation has no influence on any other

One-Tailed vs. Two-Tailed Hypotheses

Type	Null Hypothesis	Alternative Hypothesis
Two-tailed	H0: p = p0	HA: p ≠ p0
One-tailed (Left)	H0: p = p0	HA: p < p0
One-tailed (Right)	H0: p = p0	HA: p > p0

Example: Calculating the p-Value

• Suppose the observed value of the test statistic is z = 1.54. Find the p-value for each of the three possible signs in the alternative hypothesis using the standard normal table.

Section 8.3: Hypothesis Testing in Four Steps

The Four Steps in Hypothesis Testing

1. Hypothesize: State your hypotheses about the population parameter.

2. Prepare: Choose an appropriate test statistic, state and check conditions required for computations, state a significance level, and use an anonymous random sample.

3. Compute to Compare: Compare the observed value of the test statistic and compare it to what the null hypothesis would suggest. Find the p-value.

4. Interpret and Conclude: Do you reject or not reject your null hypothesis? What does this mean in the context of the data?

Example: From Chapter 7

• Suppose 34% of Canadian households own a dog. In a random sample of 200 households, 40% own a dog. Test if the percentage is higher than 34%.

• H0: p = 0.34; HA: p > 0.34

• Check conditions: n × p0 = 68, n × (1 - p0) = 132, both ≥ 10; sample is random; large population.

• Compute z and p-value; interpret and conclude.

Example: NHL Shootouts

• Test if the team shooting first in a shootout has an advantage.

• Calculate the test statistic and p-value; interpret the result in context.

Example: Increased Library Funding

• Suppose 140 out of 250 randomly selected voters favor increased funding for public libraries. Test if the majority of voters favor increased funding.

• Follow the four steps: state hypotheses, check conditions, compute test statistic and p-value, interpret and conclude.

Section 8.4: Comparing Proportions from Two Populations

Comparing Results of Two Surveys

• Suppose two surveys are conducted in different years to compare the proportion of Canadians supporting a policy.

• Let p1 and p2 represent the proportions in each year.

Changes to the Hypotheses

Comparison	Null Hypothesis	Alternative Hypothesis
Two-tailed	p1 = p2	p1 ≠ p2
One-tailed (Left)	p1 = p2	p1 < p2
One-tailed (Right)	p1 = p2	p1 > p2

Test Statistic for Two Proportions

The test statistic for comparing two proportions is:

$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}}$

• \( \hat{p}_1, \hat{p}_2 \): sample proportions from each group

• \( \hat{p} \): pooled sample proportion

• \( n_1, n_2 \): sample sizes

Conditions for Two-Proportion Test

• Large samples: n1 × p, n1 × (1-p), n2 × p, n2 × (1-p) all ≥ 10

• Random samples

• Independent samples

• Independent observations within samples

Example: Pizza Girls and Boys

• Suppose you want to test if the proportion of girls who like pizza is larger than that of boys.

• State hypotheses, check conditions, compute test statistic, use technology for p-value, and interpret the result.

Example: Doctor-Assisted Suicide

• Compare the proportion of Canadians supporting doctor-assisted suicide in two different years.

• Follow the four steps: state hypotheses, check conditions, compute test statistic and p-value, interpret and conclude.

Significance Level and Power

• Type I Error (α): Reject H0 when it is TRUE.

• Type II Error (β): Do not reject H0 when it is FALSE.

• Reducing the significance level (α) increases the probability of Type II Error (β), unless the sample size is increased.

• The only way to reduce both errors is to increase the sample size.

Statistical vs. Practical Significance

• A result is statistically significant when the null hypothesis is rejected; however, statistical significance does not necessarily mean the result is important in practice.

• A practically significant result is both statistically significant and meaningful in context.

Best Practices in Hypothesis Testing

• Never change your hypotheses after seeing the data. Hypotheses must be written before data analysis.

• When writing conclusions, avoid saying "We accept H0" or "We have proved H0 is true." Instead, say "We cannot reject H0" or "There is insufficient evidence to reject the null hypothesis."

Application Example: Treating Prostate Cancer

• Suppose an experiment compares the percentage of men who develop prostate cancer in two groups: one receiving a drug and one receiving a placebo.

• Calculate the sample percentages and test if the drug lowers the chance of getting prostate cancer using the hypothesis testing steps outlined above.

Additional info: These notes are based on lecture slides and provide a comprehensive overview of hypothesis testing for population proportions, including both one-sample and two-sample cases, with emphasis on the logic, computation, and interpretation of statistical tests.