1. 6.1 The Role of Probability in Statistics: Statistical Significance
LEARNING GOAL
Understand the concept of statistical significance and the essential role that probability plays in defining it.
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2. Definition
A set of measurements or observations in a statistical study is said to be statistically significant if it is unlikely to have occurred by chance.
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3. EXAMPLE 1 Likely Stories?
A detective in Detroit finds that 25 of the 62 guns used in crimes during the past week were sold by the same gun shop. This finding is statistically significant.
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4. EXAMPLE 1 Likely Stories?
b. In terms of the global average temperature, five of the years between 1990 and 1999 were the five hottest years in the 20th century. Having the five hottest years in 1990–1999 is statistically significant.
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5. EXAMPLE 1 Likely Stories?
c. The team with the worst win-loss record in basketball wins one game against the defending league champions.
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6. From Sample to Population
EXAMPLE 2 Statistical Significance in Experiments
A researcher conducts a double-blind experiment that tests whether a new herbal formula is effective in preventing colds. During a three-month period, the 100 randomly selected people in a treatment group take the herbal formula while the 100 randomly selected people in a control group take a placebo.
The results show that 30 people in the treatment group get colds, compared to 32 people in the control group. Can we conclude that the herbal formula is effective in preventing colds?
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7. Quantifying Statistical Significance
The definition of statistical significance that we’ve been using so far is too vague. We need a way to quantify the idea of statistical significance.
In general, we determine statistical significance by using probability to quantify the likelihood that a result may have occurred by chance. We therefore ask a question like this one:
Is the probability that the observed difference occurred by chance less than or equal to 0.05 (or 1 in 20)?
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8. Quantifying Statistical Significance
• If the probability of an observed difference occurring by chance is 0.05 (or 1 in 20) or less, the difference is statistically significant at the 0.05 level.
• If the probability of an observed difference occurring by chance is 0.01 (or 1 in 100) or less, the difference is statistically significant at the 0.01 level.
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9. EXAMPLE 2 Polio Vaccine Significance
In the test of the Salk polio vaccine (see Section 1.1), 33 of the 200,000 children in the treatment group got paralytic polio, while 115 of the 200,000 in the control group got paralytic polio. Calculations show that the probability of this difference between the groups occurring by chance is less than Describe the implications of this result.
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