8.3. Testing a Mean Hypothesis http://www.ck12.org
Next, we use our alpha level(α)of.05 and thet-distribution table to find our critical values. For a two-tailed test
with 19 degrees of freedom and a.05 level of significance, our critical values are equal to 2.093 standard errors
above and below the mean.
In calculating the test statistic, we use the formula:
t=X ̄s−x ̄μ=^3.^18 −^3.^10
0. 54 /
√
20
≈ 0. 66
This means that the observed sample mean( 3. 18 )of football players is 0.66 standard errors above the hypothesized
value of 3. 10 .Becauset= 0 .66 does not exceed 2.093 (the standard critical value), the null hypothesis is not rejected.
Therefore, we can conclude that the difference between the sample mean and the hypothesized value is not sufficient
to attribute it to anything other than sampling error. Thus, the athletic director can conclude that the mean academic
performance of football players does not differ from the mean performance of other student athletes.
How to Interpret the Results of a Hypothesis Test
In the previous section, we discussed how to interpret the results of a hypothesis test. As a reminder, when we
reject the null hypothesis we are saying that the difference between the observed sample mean and the hypothesized
population mean is too great to be attributed to chance. When we fail to reject the null hypothesis, we are saying
that the difference between the observed sample mean and the hypothesized population mean is probable if the null
hypothesis is true. Essentially, we are willing to attribute this difference to sampling error.
But what is meant bystatistical significance? Technically, the difference between the hypothesized population
mean and the sample mean is said to bestatistically significantwhen the probability that the difference occurred by
chance is less than the significance(α)level. Therefore, when the calculated test statistic (whether it is thez- or
thet-score) falls in the area beyond the critical score, we say that the difference between the sample mean and the
hypothesized population mean isstatistically significant.When the calculated test statistic falls in the area between
the critical scores we say that the difference between the sample mean and the hypothesized population mean isnot
statistically significant.
Lesson Summary
- When testing a hypothesis for the mean of a distribution, we follow a series of four basic steps:
- State the null and alternative hypotheses.
- Set the criterion (critical values) for rejecting the null hypothesis.
- Compute the test statistic.
- Decide about the null hypothesis and interpret our results.
- When we reject the null hypothesis we are saying that the difference between the observed sample mean and the
hypothesized population mean is too great to be attributed to chance. - When we fail to reject the null hypothesis, we are saying that the difference between the observed sample mean
and the hypothesized population mean is probable if the null hypothesis is true. - We use thet-distribution in hypothesis testing the same way that we use the normal distribution. However, the
t-distribution is used when the sample size is small (typically less than 120) and the population standard deviation is
unknown. - When calculating thet-statistic, we use the formula: