How do you perform a one-sample t-test?

How do you perform a one-sample t-test?

• A. t = (s − σ0) / √n
• B. t = (p̂ − p0) / √[p̂(1-p̂)/n]
• C. t = (x̄ − μ0) / (σ/√n)
• D. t = (x̄ − μ0) / (s/√n)

Answer: (D)

Testing a population mean when the population standard deviation is unknown uses the t-statistic. You compare the observed sample mean to the hypothesized mean, but you must account for sampling variability using the sample standard deviation as an estimate of σ. The standard error of the mean is s/√n, so the correct statistic is t = (x̄ − μ0) / (s/√n). This uses the t distribution with n−1 degrees of freedom, reflecting the extra uncertainty from estimating σ with s. The other formulas don’t fit: using s − σ0 in the numerator doesn’t measure how far the sample mean is from the hypothesized mean; a formula with p̂ and p0 applies to proportions, not means; and using σ in the denominator implies knowing the population standard deviation (a z-test), not the t-test.

Testing a population mean when the population standard deviation is unknown uses the t-statistic. You compare the observed sample mean to the hypothesized mean, but you must account for sampling variability using the sample standard deviation as an estimate of σ. The standard error of the mean is s/√n, so the correct statistic is t = (x̄ − μ0) / (s/√n). This uses the t distribution with n−1 degrees of freedom, reflecting the extra uncertainty from estimating σ with s.

The other formulas don’t fit: using s − σ0 in the numerator doesn’t measure how far the sample mean is from the hypothesized mean; a formula with p̂ and p0 applies to proportions, not means; and using σ in the denominator implies knowing the population standard deviation (a z-test), not the t-test.