How can a sample mean be used to test an assumption about a radio audience’s average age?

To test an assumption about a radio audience’s average age, you can use hypothesis testing, a form of statistical inference that evaluates whether sample data supports a specific claim about a population parameter.1

The process follows these steps:

• Formulate Hypotheses: Define a null hypothesis ($H_0:\mu =30$) representing the assumption that the average age is 30, and an alternative hypothesis ($H_a:\mu \ne 30$) representing the possibility that the true average differs.1
• Calculate Sample Mean: Collect a sample of individuals from the audience and compute the sample mean age, denoted as $\overline{x}$.1
• Evaluate Significance: Use the sampling distribution of $\overline{x}$ to determine if the observed mean is "close" to 30. A result that is not close provides evidence to reject $H_0$.1
• Determine Probability: Calculate the p-value, which measures the likelihood of the sample results assuming $H_0$ is true. If the p-value is less than a predetermined significance level (e.g., $\alpha =0.05$), the null hypothesis is rejected.21

If the sample size is small, a Student’s t-test may be used to perform this analysis when the population standard deviation is unknown.3