Chapter 11: Testing Hypotheses About Proportions
Recall the 5 Steps in Any Hypothesis Test
- Determine the null and alternative hypothesis
- Summarize the Data into an Appropriate Test Statistic
- Assuming the null hypothesis is true, find the p-value.
- Decide whether or not the result is statistically significant based on the p-value.
- Report the conclusion in the context of the situation.
11.1: Formulating Hypothesis Statements
Null Hypothesis () is a statement that there is nothing happening (no relationship or no difference).
Alternative Hypothesis () is a statement that something is happening (there is a relationship or there is a difference).
Examples:
Null Hypothesis:
- There is no difference between the mean height of men and women.
- There is no relationship between highest education level achieved and yearly salary.
Alternative Hypothesis:
- Men are taller than women.
- Increasing the education level achieved increases one’s yearly salary.
Ex. Suppose that a statistics teacher, we’ll call him Teacher O., asks his students to each randomly pick one of the numbers 0, 1, 2, 3, or 4. His general theory is that the proportion who pick the number 0 will be less than what would be expected from random selection. Use p to represent the population proportion that would pick the number 0.
a)What is the specific value of p that corresponds to true random selection?
p=(1/5)=0.2
b)In terms of p, write null and alternative hypotheses for this situation.
: p = 0.2
: p < 0.2
Ex. For each of the following situations, write the null and alternative hypotheses in words and in symbols.
a)In the last census, taken five years ago, it was determined that 6% of school-aged children in a certain state lived with their grandparent(s). To support a bill on tax breaks for seniors, a congressional member plans to take a random sample of school-aged children to determine if that percent has increased.
Null: The percent of children in the state who live with their grandparent(s) remains at 6% (or possibly even less).
Alternative: The percent of children in the state who live with their grandparent(s) has increased since the last census and is now greater than 6%.
: p = 0.6
: p > 0.6
**Note: The Null Hypothesis could also be written as
: p .06. (Meaning the percent has stayed the same
or decreased).
- An anthropologist is trying to determine if the people in a certain region descend from the same ancestors as another region she has studied. She knows that in the region she has studied, 15% of the people have a certain unique genetic trait. She plans to take a random sample of people in the new region and test them for that trait.
Null: In the new region being studied, 15% of the population has the unique genetic trait.
Alternative: In the new region being studied, the percent with the unique genetic trait differs from 15%.
(Notice that a two-sided alternative is used because the anthropologist did not specify a particular way in which the proportion might differ from 0.15 in the new region.)
: p = 0.15
: p 0.15
One-Sided Test: When the alternative hypothesis specifies a single direction.
Ex. p < 0.3 or p > 0.3
*Most hypothesis tests are one-sided tests because investigators usually have a particular direction in mind when they consider a question.*
Two-Sided Test (Two-Tailed Hypothesis Test): When the alternative hypothesis includes values in either direction from the test value.
Ex. p 0.3
*Used to verify that a difference exists…but does not say which direction the difference exists.*
11.2 &11.3: The Logic of Hypothesis Testing and Reaching a Conclusion about the Two Hypotheses
*Much like the criminal justice system*
i.e. You are innocent until proven guilty.
*We assume that the null hypothesis is a truth until we have sufficient evidence to conclude otherwise.
-Whenever we perform a hypothesis test we must decide to be in favor of the null or the alternative.
Test Statistic: The data summary that we used to evaluate the two hypotheses. (In Chapter 6 we used the Chi-Squared Test Statistic).
p-value: If the null hypothesis is true, the likelihood we observe a test statistic as extreme as we did. (i.e. the probability that we would see our test statistic from our sample given the null hypothesis holds for the population).
Level of Significance: A set level at which we test our p-value in order to determine statistical significance. () The level of significance is usually 0.05 or 0.01.
More on the p-value:
We can think of the p-value as a conditional probability P(A|B).
A = observing a test statistic value as extreme as that observed or more so, and
B = The null hypothesis is true.
Rejecting the Null:
When our p-value is less than our significance level, we can reject the null in favor of the alternative and conclude a statistically significant relationship exists (depending on the actual alternative that we set up).
Ex.
a) p = 0.04= 0.05
0.04<0.05 Therefore, p <
We reject the null in favor of the alternative.
b) p = 0.03= 0.01
0.03 > 0.01 Therefore, p >
We cannot reject the null hypothesis (There is not enough evidence to reject the null hypothesis).
**Again…Never Say “We Accept the Null Hypothesis”.
Ex. The following was taken from a recent journal article: “In a recent survey of hospital patients 96 out of 150 reported that they would prefer to be friends with their nurse as opposed to their doctor. If there is no difference in preference in the population, the chance of such an extreme result in a sample of this size is about 0.043. Because 0.043 is less than 0.05, we can conclude that there is a statistically significant difference in preference.” Give a numerical value for each of the following:
a) p-value
p = 0.043
b) The level of significance,
=0.05
c) The Sample Proportion
= (96/150) = 0.64
d) The Sample Size
n=150
11.4: Testing Hypothesis About a Proportion
Null Value: The specific value specified in the hypotheses for the population proportion of interest. ( )
Ex.
: p = 0.3
This implies the null value is 0.3.
: p < 0.2
This implies the null value is 0.2