Testing Hypotheses about a Population Proportion

Lecture 29

Sections 9.1 – 9.3

Tue, Oct 23, 2007

Discovering Characteristics of a Population Any question about a population must first

be described in terms of a population parameter.

We will work with the population mean and the population proportion p.

Discovering Characteristics of a Population Then the question about that parameter

generally falls into one of two categories.Estimation

What is the value of the parameter?

Hypothesis testing Does the evidence support or refute a claim about

the value of the parameter?

Examples

If we want to learn about voters’ preferences, how do we phrase the question?What parameter do we use?Do we estimate a parameter or test a

hypothesis?

Example

If we want to learn about the effectiveness of a new drug, how do we phrase the question?What parameter do we use?Do we estimate a parameter or test a

hypothesis?

Example

If we want to find out whether a newborn child is more likely to be male than female, how do we phrase the question?What parameter do we use?Do we estimate a parameter or test a

hypothesis?

Example

A standard assumption is that a newborn baby is as likely to be a boy as to be a girl. However, some people believe that boys are more likely.

Suppose a random sample of 1000 live births shows that 520 are boys and 480 are girls.

We will test the hypothesis that male births are as likely as female births, using these data.

p-Value Approach

H0

p-Value Approach

H0

p-Value Approach

H0

Observed value

p-Value Approach

0z

H0

z

Observed value

p-Value Approach

0z

Reject

p-value <

H0

z

p-Value Approach

0z

H0

Observed value

p-Value Approach

0z

H0

z

Observed value

p-Value Approach

0z

p-value >

H0

Accept

z

The Steps of Testing a Hypothesis (p-Value Approach) The seven steps:

1. State the null and alternative hypotheses.2. State the significance level.3. State the formula for the test statistic.4. Compute the value of the test statistic.5. Compute the p-value.6. Make a decision.7. State the conclusion.

The Steps of Testing a Hypothesis (p-Value Approach) See page 566. (Our seven steps are

modified from what is in the book.)

Step 1: State the Null and Alternative Hypotheses Let p = proportion of live births that are

boys. The null and alternative hypotheses are

H0: p = 0.50.

H1: p > 0.50.

State the Null and Alternative Hypotheses The null hypothesis should state a

hypothetical value p0 for the population proportion.H0: p = p0.

State the Null and Alternative Hypotheses The alternative hypothesis must contradict

the null hypothesis in one of three ways:H1: p < p0. (Direction of extreme is left.)

H1: p > p0. (Direction of extreme is right.)

H1: p p0. (Direction of extreme is left and right.)

Explaining the Data

The observation is 520 males out of 1000 births, or 52%. That is, p^ = 0.52.

Since we observed 52%, not 50%, how do we explain the discrepancy?Chance, orThe true proportion is not 50%, but something

larger, maybe 52%.

Step 2: State the Significance Level

The significance level should be given in the problem.

If it isn’t, then use = 0.05. In this example, we will use = 0.05.

The Sampling Distribution of p^

To decide whether the sample evidence is significant, we will compare the p-value to .

If p-value < , then we reject H0.

If p-value > , then we reject H0.

The Sampling Distribution of p^

We know that the sampling distribution of p^ is normal with mean p and standard deviation

Thus, under H0 we assume that p^ has mean p0 and standard deviation:

n

ppp

1ˆ

n

ppp

00ˆ

1

Step 3: The Test Statistic

Test statistic – The z-score of p^, under the assumption that H0 is true.

Thus,

npp

pppZ

p

p

00

0

ˆ

ˆ

1

ˆˆ

The Test Statistic

In our example, we compute

Therefore, the test statistic is

.01581.0

1000

50.1)50(.ˆ

p

01581.0

50.0ˆ p

Z

The Test Statistic

Now, to find the value of the test statistic, all we need to do is to collect the sample data, find p^, and substitute it into the formula for z.

Step 4: Compute the Test Statistic In the sample, p^ = 0.52. Thus,

265.101581.0

50.052.0

Z

Step 5: Compute the p-value

To compute the p-value, we must first check whether it is a one-tailed or a two-tailed test.

We will compute the probability that Z would be at least as extreme as the value of our test statistic.

If the test is two-tailed, then we must take into account both tails of the distribution to get the p-value. (Double the value in one tail.)

Compute the p-value

In this example, the test is one-tailed, with the direction of extreme to the right.

So we compute

p-value = P(Z > 1.265) = 0.1029.

Compute the p-value

To find this value, we evaluate

normalcdf(0.52, E99, 0.50, 0.01581)

on the TI-83.

Step 6: Make a Decision

Since the p-value is greater than , our decision is: Do not reject the null hypothesis.

The decision is stated in statistical jargon.

Step 7: State the Conclusion

State the conclusion in a sentence: It is not true that more than 50% of live births

are male. The conclusion must state the decision in

the language of the original problem. It should not use statistical jargon.

Summary

1. H0: p = 0.50

H1: p > 0.502. = 0.05.3. Test statistic:

4. z = (0.52 – 0.50)/0.0158 = 1.26.5. p-value = P(Z > 1.26) = 0.1038.

6. Do not reject H0.7. It is not true that more than 50% of live births are male.

n

pp

ppZ

00

0

1

ˆ

Beforecollecting

data

Summary

1. H0: p = 0.50

H1: p > 0.502. = 0.05.3. Test statistic:

4. z = (0.52 – 0.50)/0.0158 = 1.26.5. p-value = P(Z > 1.26) = 0.1038.

6. Do not reject H0.7. It is not true that more than 50% of live births are male.

n

pp

ppZ

00

0

1

ˆ

Aftercollecting

data

Summary

1. H0: p = 0.50

H1: p > 0.502. = 0.05.3. Test statistic:

4. z = (0.52 – 0.50)/0.0158 = 1.26.5. p-value = P(Z > 1.26) = 0.1038.

6. Do not reject H0.7. It is not true that more than 50% of live births are male.

n

pp

ppZ

00

0

1

ˆ