Construct your null and alternative hypotheses.
The null hypothesis, Ho, is always the opposite of what you are testing for.
Ho: The sample mean is not significantly different from the population mean.
The alternative hypothesis, Ha, will vary depending on whether the test is one-tailed or two-tailed. If you are testing to determine if the sample mean is different from, or equal to, the population mean, the test is two-tailed. If you are testing to determine if the sample mean is either greater than or less than the population mean, the test is one-tailed.
For a one-tailed test, Ha: The sample mean is significantly greater/lower than the population mean.
For a two-tailed test, Ha: The sample mean is significantly different from the population mean.
Subtract the population mean from the mean of the sample that you wish to hypothesis test.
Calculate your standard error by dividing the population standard deviation by the square root of the n from your sample.
Calculate your the Z-score by dividing the result of step 2 by the standard error found in step 3.
Compare the Z-score to the Z table in the resources section. The numbers on the left represent the first two numbers, while the numbers on the top represent the third number. Go to the box where the row with your first two numbers intersects the column with your third number. We will use this number to calculate our p-value.
For a lower tailed test, Ha: The sample mean is significantly lower than the population mean. The number found in the box is your p-value:
For Z = -1.91 p = .0281
For an upper tailed test, Ha: The sample mean is significantly greater than the population mean. The p-value is 1 minus the number in the box:
For Z = 1.91 p = (1 - .9719) = .0281
For a two-tailed test, Ha: The sample mean is different from the population mean. The p-value is double the value found for a one-tailed test:
For Z = 1.91 p = (.0281 X 2) = .0562
Find alpha from your confidence level. Alpha = (100 - confidence level)/100.
For a confidence level of 95%, alpha = (100 - 95)/100 = .05
For a confidence level of 98%, alpha = (100 - 98)/100 = .02
Compare your p-value to alpha. Discover if your p-value is greater than, less than, or equal to alpha.
Draw a conclusion. If your p-value is less than alpha, reject the null hypothesis and accept the alternative hypothesis:p = .0281 alpha = .05; .0281 < .05 Therefore, we reject the null hypothesis and conclude, with 95% confidence, that the sample mean is significantly greater/less/different from the population mean. If your p-value is greater than or equal to alpha, you must fail to reject the null hypothesis: p = .0562 alpha = .05; .0562 > .05 Therefore, we fail to reject the null hypothesis and conclude that the sample mean is not significantly different from the population mean.