LOS G and H requires us to:
g. identify the appropriate test statistic and interpret the results for a hypothesis test concerning the population mean of both large and small samples when the population is normally or approximately normally distributed and the variance is 1) known or 2) unknown
h. identify the appropriate test statistic and interpret the results for a hypothesis test concerning the equality of the population means of two at least approximately normally distributed populations, based on independent random samples with 1) equal or 2) unequal assumed variances
In most cases, the variance of the underlying population is either not known or cannot be calculated. In all such cases, the test of a single mean is either a t-test or a z-test.
1. t-Test
a. The t-test is done to compare a single mean with a value.
b. The t-test can be used with a degree of freedom of n-1, when:
i. Population variance is unknown and
ii. The sample is large or sample is small but (approximately) normally distributed.
c. The test statistic for the t-test is:
2. z-Test
a. z-Test is done to compare a single mean with a value.
b. The t-test can be used, when:
i. Sample size is large, or
ii. the population is normally distributed.
c. The z-test can be used whether the population variance is known or unknown.
d. The test statistic for z-test when the population variance is known is:
e. And, the test statistic for the z-test when the population variance is unknown is: