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# Test of Significance involving the Sample Average

A bowler is bragging that his average is at least 180''. We observe him play three games, his scores are 125, 155, 140 ( , S=15). Should we accept or reject his claim? We should reject it. Why? Because a sample average as low as 140 is unlikely from a 180 bowler. How unlikely? A 180 bowler will bowl a 3-game average of 140 or lower only 2 percent of the time. Is 2 percent of the time unlikely? In statistics, yes. 5 percent or less is called statistically significant.

The decision making process above is called a test of significance   . Here is the way a statistical report would formally present the test, in numbered stages.

1. Hypotheses: versus
2. Test Statistic:
3. P-value: Presuming H0 is true, the likelihood of chance variation yielding a t-statistic as low as -4.62 is .02. (Calculation details later.)

4. Conclusion: Since P-value , the observed sample value is declared significantly unlikely under . Hence, we reject H0 and conclude . The sample provides evidence to reject the bowler's claim.

Here is a more detailed description of each component of the test of significance above.

1.
The null and alternative hypotheses    .

H0  and H1  are called the null hypothesis  and alternative hypothesis , respectively. The two hypotheses describe the two possibilities: the claim is true ( ), or the claim is false (). Note that

(i) the two hypotheses are statements about the population (ii) the two hypotheses are complementary; if one occurs the other does not (iii) the hypothesis with the equal sign is the null hypothesis
A test of significance rejects (population statement) H0 and concludes H1 if the sample values are significantly far from H0 and inside H1''. Hence, we reject and conclude if is some significant distance below 180. How far below 180 is significant''? The test statistic helps us determine where to draw the line in the sand.

2.
The Test Statistic For tests of hypotheses on , the t-test statistic  is a ratio of the form

For the null hypothesis , the t-test statistic is

H0 will be rejected if and only if will be some significant distance below 180, which happens if and only if t is some significant distance below 0. Based on the sample observed scores, the observed t value is

Is t=-4.62 significantly below 0?'' To answer this, we will need the help of the t-curve with n-1 degrees of freedom.

3.
The P-value

Using the t curve with n-1=2 degrees of freedom, the likelihood of chance variation resulting in a t-value as low as -4.62 is .02.

Since this likelihood is less than .05 (the standard for statistical significance), we declare that t=-4.62 is significantly below 0'', or that  is significantly below 180'', and reject . In general, the P-value is the total area under the curve more extreme than t in support of H1. If t is deep in H1 territory'', then the P-value is small. If P-value .05, we reject H0 with statistical significance. If P-value .01, we reject H0 with high statistical significance. If P-value is larger than .05, we accept H0.

4.
Conclusion If H0 is rejected, the conclusion is usually stated as there is enough evidence to ...' or there are statistically significant differences...'. If H0 is accepted, the conclusion is usually stated as there is not enough evidence to ...', or there are no statistically significant differences...'. Since P-value=.02 in our example, we conclude that the sample provides enough evidence to reject the bowler's claim of a 180 average'. Or his performance ( ) was much lower than his claimed average (), and the difference is statistically significant.'

Summary of the lower-tailed  t-test for :

versus
Test statistic:
P-value: Total area less than t (the direction of H1) under t-curve with n-1 degrees of freedom If t is significantly below 0 (the direction of H1), the P-value will be small.
Conclusion: If P-value .05, we reject H0 with statistical significance. If P-value .01, we reject H0 with high statistical significance. If P-value >.05, we do not reject H0.

Next: Upper-tailed t-test Up: Testing Hypotheses Previous: Testing Hypotheses

2003-09-08