3 if we reject the null hypothesis then we can assume

3. If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
4. Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on
probability laws; therefore, we can talk only in terms of non-absolute certainties.
9.2 Outcomes and the Type I and Type II Errors
In every hypothesis test, the outcomes are dependent on a correct interpretation of the data. Incorrect calculations or
misunderstood summary statistics can yield errors that affect the results. A Type I error occurs when a true null hypothesis
is rejected. A Type II error occurs when a false null hypothesis is not rejected.
The probabilities of these errors are denoted by the Greek letters α and β , for a Type I and a Type II error respectively. The
power of the test, 1 – β , quantifies the likelihood that a test will yield the correct result of a true alternative hypothesis being
accepted. A high power is desirable.
9.3 Distribution Needed for Hypothesis Testing
In order for a hypothesis test’s results to be generalized to a population, certain requirements must be satisfied.
When testing for a single population mean:
1. A Student's t -test should be used if the data come from a simple, random sample and the population is approximately
normally distributed, or the sample size is large, with an unknown standard deviation.
2. The normal test will work if the data come from a simple, random sample and the population is approximately
normally distributed, or the sample size is large.
When testing a single population proportion use a normal test for a single population proportion if the data comes from a
simple, random sample, fill the requirements for a binomial distribution, and the mean number of success and the mean
number of failures satisfy the conditions: np > 5 and nq > n where n is the sample size, p is the probability of a success, and
q is the probability of a failure.
9.4 Full Hypothesis Test Examples
The hypothesis test itself has an established process. This can be summarized as follows:
1. Determine H
0
and H
a
. Remember, they are contradictory.
2. Determine the random variable.
3. Determine the distribution for the test.
4. Draw a graph, calculate the test statistic, and if you choose then, use the test statistic to calculate the p -value. (A z -
score and a t -score are examples of test statistics.)
5. Compare the calculated test statistic with the Z critical value determined by the level of confidence required by the
test and make a decision (cannot reject H
0
or cannot accept H
0
). Alternatively compare the preconceived α with the
p -value, make a decision (cannot reject H
0
or cannot accept H
0
), and write a clear conclusion using English sentences.
9.5 Rare Events, the Sample, Decision and Conclusion
Chapter 9 | Hypothesis Testing with One Sample 383