9.11: Testing a Claim about Mean: Known Population SD

A complete procedure of testing the hypothesis about a population mean is explained here.

Estimating a population mean requires the samples to be distributed normally. The data should be collected from the randomly selected samples having no sampling bias. The sample size needed to be higher than 30, and most importantly, the population standard deviation should be already known.

In most realistic situations, the population standard deviation is often unknown, but in rare circumstances, when it is known, the claim about the population mean can be tested easily using the normality assumption and 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 requires normal distribution, the critical value is calculated using the z distribution (z table). It is calculated at the desired confidence level, most commonly at 95% or 99%. As per the traditional method, the z statistic calculated from the sample data is compared with the z score. The P-value is calculated based on the data as per the P-value method. Both these methods help conclude the hypothesis test.

Exposure to different light wavelengths may affect the spawning rate in zebrafish.

So, an experiment is conducted where one group of 50 zebrafish is exposed to blue light, and their spawning rate is compared with the control group having the same sample size.

To test the claim, we begin with the null hypothesis that the mean spawning rate in the exposed and the control group is the same and an alternative hypothesis that the blue light increases the mean spawning rate.

The experiment showed that the mean spawning rate in the exposed group was 550 per fish, whereas, for the control group, it was 250.

Calculating the test statistic from these data requires prior knowledge of population standard deviation, which is 146, known from the previous studies.

Using these data, we can calculate the z statistic and observe that it falls in the critical region at the significance level of 0.05.

Additionally, the P-value for this z statistic is less than 0.05, concluding that the blue light enhances the spawning rate in zebrafish.