Additional Information and Full Hypothesis Test Examples
Additional Information and Full Hypothesis Test Examples
Learning Outcomes
- Conduct and interpret hypothesis tests for a single population proportion.
- In a hypothesis test problem, you may see words such as “the level of significance is 1%.” The “1%” is the preconceived or preset α.
- The statistician setting up the hypothesis test selects the value of α to use before collecting the sample data.
- If no level of significance is given, a common standard to use is α = 0.05.
- When you calculate the p-value and draw the picture, the p-value is the area in the left tail, the right tail, or split evenly between the two tails. For this reason, we call the hypothesis test left, right, or two tailed.
- The alternative hypothesis, Ha, tells you if the test is left, right, or two-tailed. It is the key to conducting the appropriate test.
- Ha never has a symbol that contains an equal sign.
- Thinking about the meaning of the p-value: A data analyst (and anyone else) should have more confidence that he made the correct decision to reject the null hypothesis with a smaller p-value (for example, 0.001 as opposed to 0.04) even if using the 0.05 level for alpha. Similarly, for a large p-value such as 0.4, as opposed to a p-value of 0.056 (alpha = 0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than mindlessly applying rules.
The following examples illustrate a left-, right-, and two-tailed test.
Example
Ho: μ = 5, Ha: μ < 5
Test of a single population mean. Ha tells you the test is left-tailed. The picture of the p-value is as follows:
TRY IT
H0: μ = 10, Ha: μ < 10
Assume the p-value is 0.0935. What type of test is this? Draw the picture of the p-value.
Solution:
left-tailed test
Example
H0: p ≤ 0.2 Ha: p > 0.2
This is a test of a single population proportion. Ha tells you the test is right-tailed. The picture of the p-value is as follows:
TRY IT
H0: μ ≤ 1, Ha: μ > 1
Assume the p-value is 0.1243. What type of test is this? Draw the picture of the p-value.
Solution:
right-tailed test
Example
This is a test of a single population mean. Ha tells you the test is two-tailed. The picture of the p-value is as follows.
TRY IT
H0: p = 0.5, Ha: p ≠ 0.5
Assume the p-value is 0.2564. What type of test is this? Draw the picture of the p-value.
Solution:
two-tailed test
Full Hypothesis Test Examples
Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds. His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for 15 25-yard freestyle swims. For the 15 swims,Jeffrey’s mean time was 16 seconds. Frank thought that the goggles helped Jeffrey to swim faster than the 16.43 seconds. Conduct a hypothesis test using a preset α = 0.05. As