In this chapter we introduce the idea of hypothesis testing in general, and then we look at the specifics for a hypothesis test for a single population proportion.

Chapter 17Hypothesis Tests for a Single Proportion

The Idea

The idea of hypothesis testing is to use a single, properly collected sample to test whether or not an assumed value for some population parameter is actually true.

We will make use of the sampling distributions that we have been using in the last couple of chapters.

The Hypotheses

The null hypothesis, denoted H0, is a statement about a population parameter that is assumed to be true.

i.e. = #, p = #, = #, etc…

The alternative hypothesis, denoted Ha, is a statement that in some way the null hypothesis is not correct. The “=” is replaced by “>” or “<” or “≠”.

This is the question that is under investigation.

Example 1

Write the hypotheses under investigation for each of the following scenarios.

(a) The Mars Candy Co. claims that since the introduction of blue as a color for M&M’s, 15% of all M&M’s produced are blue. In a randomly collected sample of 80 M&M’s, only 11 were blue. Is this strong enough evidence to claim that the true proportion of all M&M’s that are blue is less than 15%?

Example 1continued

Write the hypotheses under investigation for each of the following scenarios.

(b) In October 2000, the results of a large survey showed that 84.9% of 12460 males and 88.1% of 12678 females had graduated from high school. Does this study support the claim that females were more likely to graduate from high school than males in 2000?

Example 1continued

Write the hypotheses under investigation for each of the following scenarios.

(c) The posted speed limit on a certain residential road is 30mph. The residents believe that drivers are speeding on this road on average. They observe 20 randomly selected drivers on this road and find the mean speed to be 31.8mph with a standard deviation of 4.2mph. Is the residents’ belief accurate?

Example 1continued

Write the hypotheses under investigation for each of the following scenarios.

(d) Are the prices charged for a used camera higher on average when buying from a stranger than when buying from a friend?

Example 1continued

Write the hypotheses under investigation for each of the following scenarios.

(e) Mars Candy Co. claims that of all its peanut M&M’s, 20% are yellow, 20% are red, 10% each of orange, blue, and green, and the remaining were brown. Does the following data from a randomly selected sample support this claim?

Test Statistic and P-value

The test statistic for a sample is a numerical summary of the data-set. It is a single numerical value on some probability curve (the z-curve, one of the Gosset t-curves, the curve, etc…) that is representative of the sample data.

The P-value is the probability of getting a sample with a similar test statistic as the one we have collected under the assumption that the null hypothesis (H0) is true. That is, it is the probability of getting a sample similar to the one we have.

P-value and Conclusions

A large P-value would say that, assuming H0 is true, the sample is not that unusual. This would not support the alternative hypothesis, and so our conclusion would be “fail to reject H0”.

A small P-value would say that, assuming H0 is true, the sample is unusual. Such a sample would lend support to the alternative hypothesis. Thus our conclusion would be “reject H0”.

P-value and Conclusions

How large a P-value is “too large”?

How small a P-value is “small enough”?

 

A level of significance, denoted , is chosen for the test.

If the P-value is < , we reject H0

If the P-value is > , we fail to reject H0

The default value for the level of significance is = 0.05.

Other common values are = 0.10 and = 0.01.

Tests for a Single p

Hypotheses

H0: p = #

Ha: one of (a) p > # (upper tail test)

(b) p < # (lower tail test)

(c) p ≠ # (two-tail test)

Tests for a Single p

Test Statistic

where p is the hypothesized value in H0

Tests for a Single p

P-value

Depends on the alternative hypothesis:

(a) (upper tail test)

(b) (lower tail test)

(c) (two-tailed test)

Tests for a Single p

Validity/Assumptions

• We have a properly collected, random sample

• Sample size is not more than 10% of the population

• large sample size:

Example 2

The Mars Candy Co. claims that since the introduction of blue as a color for M&M’s, 15% of all M&M’s produced are blue. In a randomly collected sample of 80 M&M’s, only 11 were blue. Is this strong enough evidence to claim that the true proportion of all M&M’s that are blue is less than 15%?

Tests for a Single pvia TI 83/84

Press , go over to “Tests”, then scroll down and

select “1-PropZTest…”

Example 2 via the calculator

Example 3

The drug Prevnar is a vaccine meant to prevent meningitis. In clinical trials, the vaccine was given to 710 randomly selected children 12 – 15 months in age. Of these, 96 experienced appetite loss as a side effect. Is this significant evidence that more than 10% of all children (age 12 – 15 months) that take Prevnar will have loss of appetite as a side affect? Test the relevant hypotheses at the = 0.05 level.

Tests for a Single pConfidence Interval Approach

We can replace the test statistic and P-value with a confidence interval for p calculated from the sample.

If the hypothesized value is not in the interval, then we reject H0

If the hypothesized value is in the interval, then we fail to reject H0

All other “pieces” of the hypothesis test are the same.

Example 4

The drug Prevnar is a vaccine meant to prevent meningitis. In clinical trials, the vaccine was given to 710 randomly selected children 12 – 15 months in age. Of these, 96 experienced appetite loss as a side effect. Is this significant evidence that more than 10% of all children (age 12 – 15 months) that take Prevnar will have loss of appetite as a side affect? Test the relevant hypotheses using a 95% confidence interval.