Maths: Testing a Population Mean with a t-Test

Hypothesis testing is the statistical process of using sample data to make a decision about a population parameter, and in this case, we are testing a claim about a population mean. When the population standard deviation is unknown and the sample size is small, we use a t-test rather than a z-test, which is exactly the situation presented here with the factory’s metal rods. The core idea is to set up two competing statements: the null hypothesis (H₀) represents the status quo—here, that the mean breaking strength equals 450 N—while the alternative hypothesis (H₁) captures the suspicion, that the mean is less than 450 N. The test statistic, t = (sample mean − hypothesized mean) / (sample standard deviation / √n), measures how many standard errors the sample mean falls from the claimed value. A negative t-value indicates the sample mean is below the hypothesized mean. The decision to reject H₀ depends on comparing this calculated t to a critical value from the t-distribution, which accounts for the small sample size through degrees of freedom (n − 1). If the calculated t is more extreme than the critical value, we reject H₀ and conclude there is sufficient evidence to support the suspicion—though we never “prove” the alternative, only that the data are inconsistent with the null.