Testing a Claim about Mean: Unknown Population SD

The t distribution is used when the population standard deviation is unknown, which is the typical real-world scenario. When sample size is less than 30 and the population standard deviation is unknown, the sample standard deviation replaces the population parameter. The t distribution accounts for this additional uncertainty, making it more appropriate than the z distribution for calculating the test statistic and critical values.

The null hypothesis is a neutral statement asserting that the population mean equals a specific value, representing no difference or effect. The alternative hypothesis expresses the original claim using an inequality sign, such as greater than, less than, or not equal to. In the lizard example, the null hypothesis states both species have equal flapping rates, while the alternative claims the pale-colored species flaps faster.

Sample size determines which distribution to use for testing. If the sample size is less than 30 and the population standard deviation is unknown, the t distribution is used. For larger samples or when population standard deviation is known, different approaches apply. The degrees of freedom for the t distribution are calculated directly from the sample size, affecting the critical value obtained from the t table.

A P-value less than 0.05 indicates strong evidence against the null hypothesis at the 95% confidence level. When the P-value is below this significance threshold, the result is considered statistically significant, supporting the alternative hypothesis. In the lizard study, the P-value from the calculated t statistic was less than 0.05, confirming that the pale-colored species flaps its dewlap significantly faster than the tricolored species.

Sample data must be collected from randomly selected samples with no sampling bias to ensure validity. The samples should be approximately normally distributed for accurate results. While there is no specific minimum sample size requirement, these conditions ensure the hypothesis test produces reliable conclusions about the population mean.

The test statistic is calculated using the sample standard deviation instead of the population standard deviation. This calculated t statistic is then compared with the critical value from the t distribution table, which is determined at specific degrees of freedom based on sample size. The test statistic's position relative to the critical region determines whether to reject the null hypothesis.

The type of hypothesis test is determined by the inequality sign used in the alternative hypothesis. A greater-than sign indicates a right-tailed test, a less-than sign indicates a left-tailed test, and a not-equal-to sign indicates a two-tailed test. In the lizard example, the alternative hypothesis uses a greater-than sign, making it a right-tailed test with the critical region at the right tail of the distribution.