Testing a Claim about Mean: Known Population SD

When population standard deviation is known, hypothesis testing about the mean becomes straightforward using the z distribution and normality assumption. Known standard deviation allows direct calculation of the z statistic from sample data, enabling comparison with critical values and significance levels. This method is rare in practice but highly efficient when the parameter is available from prior studies.

Sample data must be collected from randomly selected samples with no sampling bias, and the sample size must exceed 30 to satisfy normality assumptions. The population standard deviation must be known beforehand. These conditions ensure the z distribution applies reliably and the test statistic accurately reflects the population parameter.

The null hypothesis states that the population mean equals a specific value, representing no effect or change. The alternative hypothesis uses an inequality sign to claim the mean differs from that value. The direction of the inequality determines whether the test is right-tailed, left-tailed, or two-tailed, guiding the critical region placement.

When the calculated z statistic falls in the critical region at a chosen significance level like 0.05, it indicates the sample data provides strong evidence against the null hypothesis. This outcome leads to rejecting the null hypothesis and supporting the alternative hypothesis, suggesting a statistically significant effect exists.

The traditional method compares the calculated z statistic directly with the critical z score from the z table at a specified confidence level. The P-value method calculates the probability of observing sample data as extreme as or more extreme than what was observed, assuming the null hypothesis is true. Both methods lead to the same conclusion about rejecting or failing to reject the null hypothesis.

The z statistic calculated from the sample means (550 versus 250) and known population standard deviation of 146 fell within the critical region at the 0.05 significance level. Additionally, the P-value was less than 0.05, providing strong statistical evidence that blue light increases mean spawning rate compared to the control group.

Most realistic situations involve unknown population standard deviation, requiring the t distribution instead of the z distribution. When population standard deviation is unavailable from prior research, you must estimate it from sample data. This approach is more common in practice than testing with known standard deviation.