9.12: Testing a Claim about Mean: Unknown Population SD

A complete procedure of testing a hypothesis about a population mean when the population standard deviation is unknown is explained here.

Estimating a population mean requires the samples to be approximately normally distributed. The data should be collected from the randomly selected samples having no sampling bias. There is no specific requirement for sample size. But if the sample size is less than 30, and we don't know the population standard deviation, a different approach is used; instead of the z distribution, the t distribution is used for calculating the test statistic and critical value.

As in most realistic situations, the population standard deviation is often unknown; testing the claim about the population mean would utilize the sample standard deviation. The critical value is calculated using the t distribution (at specific degrees of freedom calculated from sample size) instead of the z distribution.

The hypothesis (null and alternative) should be stated clearly and then expressed symbolically. The null hypothesis is a neutral statement stating population mean is equal to some definite value. The alternative hypothesis can be based on the mean claimed in the hypothesis with an inequality sign. The right-tailed, left-tailed, or two-tailed hypothesis test can be decided based on the sign used in the alternative hypothesis.

As the method does not require normal distribution, the critical value is calculated using the t distribution (t table). It is generally calculated at 95% or 99% of the desired confidence level. As per the traditional method, the sample t statistic calculated from the sample data is compared with the t score (t critical value) obtained from the t table. The P-value is calculated based on the data as per the P-value method. Both these methods help conclude the hypothesis test.

Males of two species of Fan-throated lizards flap their dewlaps differently: the pale-colored species flaps its dewlap at a faster rate than the tricolored species.

Field observations of 32 individuals show that the pale-colored species flaps the dewlap on an average of 10 times per unit time while the tricolored species flaps it on an average of 6 times per unit time.

To test the claim, we begin with the null hypothesis that there is no difference between the dewlap flapping rate in both species. In contrast, the alternative hypothesis states that the pale-colored species flaps the dewlap at a faster rate, which is the original claim.

As the population standard deviation of the dewlap flapping rate is unknown, this hypothesis test is conducted using the t distribution for which the sample standard deviation is utilized to calculate the test statistic.

Here, the test statistic can be seen within the critical region at the right tail.

Additionally, the P-value from the calculated t statistic is less than 0.05 supporting the original claim.