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Hypothesis Testing of Single Mean and Single Proportion: Additional Information

Module by: Dr. Barbara Illowsky, Susan Dean

  • In a hypothesis test problem, you may see words such as "the level of significance is 1%." The "1%" is the preconceived αα.
  • The statistician setting up the hypothesis test selects the value of αα to use before collecting the sample data.
  • If no level of significance is given, we generally can use α=0.05α=0.05.
  • When you calculate the p-value and draw the picture, the p-value is 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 alternate hypothesis, HaHa, tells you if the test is left, right, or two-tailed. It is the key to conducting the appropriate test.
  • HaHa never has a symbol that contains an equal sign.

The following examples illustrate a left, right, and two-tailed test.

Example 1

H o H o : μμ = 5 =5 H a H a : μμ <5 5

Test of a single population mean. HaHa tells you the test is left-tailed. The picture of the p-value is as follows:

Normal distribution curve of a single population mean with a value of 5 on the x-axis and the p-value points to the area on the left tail of the curve.

Example 2

H o H o : pp ≤ 0.2 ≤0.2 H a H a : pp > 0.2 >0.2

This is a test of a single population proportion. HaHa tells you the test is right-tailed. The picture of the p-value is as follows:

Normal distribution curve of a single population proportion with the value of 0.2 on the x-axis. The p-value points to the area on the right tail of the curve.

Example 3

H o H o : μμ = 50 =50 H a H a : μμ ≠ 50 ≠50

This is a test of a single population mean. HaHa tells you the test is two-tailed. The picture of the p-value is as follows.

Normal distribution curve of a single population mean with a value of 50 on the x-axis. The p-value formulas, 1/2(p-value), for a two-tailed test is shown for the areas on the left and right tails of the curve.

Glossary

Hypothesis Testing:
Based on sample evidence procedure to determine whether the hypothesis stated is a reasonable statement and cannot be rejected, or is unreasonable and should be rejected.
p-value:
The probability that event will happen purely by chance assuming the null hypothesis is true. The smaller p-value, the stronger the evidence is against the null hypothesis.

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