Chapter 22

What Is a Test of Significance?

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Thought Question 1Suppose 60% (0.60) of the population are in favor of new tax legislation. A random sample of 265 people results in 175, or 0.66, who are in favor. From the Rule for Sample Proportions, we know the potential sample proportions in this situation follow an approximately normal distribution, with a mean of 0.60 and a standard deviation of 0.03. Find the standardized score for the observed value of 0.66, then find the probability of observing a standardized score at least that large or larger.

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Thought Question 1: Bell-Shaped Curve of Sample Proportions (n=265)

0.60 0.630.57 0.660.54 0.690.51

2.27%

mean = 0.60

S.D. = 0.032.0

0.03

0.600.66z

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Thought Question 2Suppose that in the previous question we do not know for sure that the proportion of the population who favor the new tax legislation is 60%. Instead, this is just the claim of a politician. From the data collected, we have discovered that if the claim is true, then the sample proportion observed falls at the 97.73 percentile (about the 98th percentile) of possible sample proportions for that sample size. Should we believe the claim and conclude that we just observed strange data, or should we reject the claim?What if the result fell at the 85th percentile?At the 99.99th percentile?

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Thought Question 2: Bell-Shaped Curve of Sample Proportions (n=265)

0.60 0.63 0.66 0.690.54 0.570.51

85th

98th

99.99th

Why do we need significance tests?

“Significance” in the statistical sense does not mean “important”. It means simply “not likely to happen just by chance”.

An outcome is very unlikely if it provides good evidence that a claim is not true. (If we took many samples and the claim were true, we would rarely get a result like this).

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Is the coffee fresh?

People of taste are supposed to prefer fresh-brewed coffee to the instant variety. But perhaps many coffee drinkers just need their caffeine fix. A skeptic claims that coffee drinkers can’t tell the difference. Let’s do an experiment to test this claim.

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Is the coffee fresh?

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Is the coffee fresh?

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Is the coffee fresh?

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Is the coffee fresh?

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Figure 22.2 The sampling distribution of the proportion of 50 coffee drinkers whoprefer fresh-brewed coffee if the truth about all coffee drinkers is that 50% preferfresh coffee. The shaded area is the probability that the sample proportion is 56% or greater.

Is the coffee fresh?

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Is the coffee fresh?

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Statistical significanceIf the P-value is as small or smaller than α, we say that the data are statistically significant at level α.

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The Five Steps to Significance Testing

• Determine the two hypotheses.• Compute the sampling distribution based on the null

hypothesis.• Collect and Summarize the data.

(Calculate the observed test statistic.)

• Determine how unlikely the test statistic is if the null hypothesis is true. (Calculate the P-value.)

• Make a decision/draw a conclusion.(based on the p-value, is the result statistically significant?)

Count Buffon’s coin

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Figure 22.3 The sampling distribution of the proportion of heads in 4040 tossesof a balanced coin. Count Buffon’s result, proportion 0.507 heads, is marked.

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Figure 22.4 The P-value for testing whether Count Buffon’s coin was balanced. This is the probability, calculated assuming a balanced coin, of a sample proportion as far or farther from 0.5 as Buffon’s result of 0.507.

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22.13 Unemployment

The national unemployment rate in a recent month was 4.9%. You think the rate may be different in your city, so you plan a sample survey that will ask the same questions as the Current Population Survey. To see if the local rate differs significantly from 4.9%, what hypotheses will you test?

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Answer for 22.13

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22.16 Using the Internet

In 2006, 75.9% of first-year college students responding to a national survey said that they used the Internet frequently for research or homework. A state university finds that 130 of an SRS of 200 of its first-year students said that they used the Internet frequently for research or homework. We wonder if the proportion at this university differs from the national value, 75.9%

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Answer for 22.16

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22.26 Do chemists have more girls?

Some people think that chemists are more likely than other parents to have female children. The Washington State Department of Health lists the parents’ occupations on birth certificates. Between 1980 and 1990, 555 children were born to fathers who were chemists. Of these births, 273 were girls. During this period, 48.8% of all births in Washington State were girls. Is there evidence that proportion of girls born to chemists is higher than the state proportion?

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Answer for 22.26

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22.27 Speeding

It often appears that most drivers on the road are driving faster than the posted speed limit. Situations differ, of course, but here is one set of data. Researchers studied the behavior of drivers on a rural interstate highway in Maryland where the speed limit was 55 miles per hour. They measured speed with an electronic device hidden in the pavement and, to eliminate large trucks, considered only vehicles less than 20 feet long. They found that 5690 out of 12931 vehicles were exceeding the speed limits. Is this good evidence that fewer than half of all drivers are speeding?

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Answer for 22.27

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