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 ( $\overline{X}=140$, 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: $H_0: \mu\geq 180$ versus $H_1: \mu < 180$

2. Test Statistic: $t = \frac{\overline{X}-180}{S/\sqrt{n}} = \frac{ 140 -180}{15/\sqrt{3}} = -4.62$

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 $\leq .05$, the observed sample value $\overline{X}=140$ is declared significantly unlikely under $H_0: \mu\geq 180$. Hence, we reject H₀ and conclude $H_1: \mu < 180$. 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 ( $\mu \geq 180$), or the claim is false ($\mu < 180$). 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 H<sub>0</sub> and inside H<sub>1</sub>''. Hence, we reject $H_0: \mu\geq 180$ and conclude $H_1: \mu < 180$ if $\overline{X}$ 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 $\mu$, the t-test statistic  is a ratio of the form

$$t = \frac{\mbox{Observed Sample Mean - $H_0$\space Value}}{\m... ...an}} = \frac{\overline{X} - H_0 \mbox{ Value}}{S/\sqrt{n}} .$$

For the null hypothesis $H_0: \mu\geq 180$, the t-test statistic is

$$t = \frac{\overline{X} - 180}{S/\sqrt{n}}$$

H₀ will be rejected if and only if $\overline{X}$ 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

$$t = \frac{\overline{X} - 180}{S/\sqrt{n}} = \frac{140 - 180}{15/\sqrt{3}} = -4.62$$

``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 `` $\overline{X}=140$ is significantly below 180'', and reject $H_0: \mu\geq 180$. 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 $\leq$ .05, we reject H₀ with statistical significance. If P-value $\leq$ .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 ( $\overline{X}=140$) was much lower than his claimed average ($\mu=180$), and the difference is statistically significant.'

Summary of the lower-tailed  t-test for $\mu$:

$H_0: \mu \geq c$ versus $H_1: \mu < c$

Test statistic: $t= (\overline{X}- c)/(S/\sqrt{n})$

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 $\leq$ .05, we reject H₀ with statistical significance. If P-value $\leq$ .01, we reject H₀ with high statistical significance. If P-value >.05, we do not reject H₀.