Z-Tests and T-Tests

Both tests compare a sample mean with a hypothesized population mean. The only difference is whether the population standard deviation is known.

z = (x̄ − μ₀) / (σ / √n)

t = (x̄ − μ₀) / (s / √n)

• x̄: the sample mean.
• μ₀: the population mean claimed by H₀.
• σ: the known population standard deviation, used in the z-test.
• s: the sample standard deviation, used in the t-test when σ is unknown.
• n: the sample size.
• Denominator: the standard error, so the result is the distance between x̄ and μ₀ measured in standard errors.

The t statistic follows a t-distribution with n − 1 degrees of freedom. For small samples, this distribution has heavier tails than the normal curve, making the test more conservative. As n grows, it becomes almost identical to the normal curve.

Example: A battery is advertised to last 50 hours. A sample of 16 batteries has a mean life of 48 hours and s = 8 hours. Test H₀: μ = 50 against H₁: μ ≠ 50 at α = 0.05.

1. SE = 8 / √16 = 2
2. t = (48 − 50) / 2 = −1
3. Degrees of freedom = 16 − 1 = 15, and the critical values for a two-tailed test at 0.05 are ±2.131.
4. Since −1 lies between −2.131 and 2.131, we fail to reject H₀.

from scipy import stats

t = -1
p = 2 * stats.t.sf(abs(t), df=15)
print(p)

Code Explanation:

• t holds the test statistic calculated above.
• stats.t.sf(abs(t), df=15) gives the area in one tail beyond |t| for a t-distribution with 15 degrees of freedom.
• Multiplying by 2 adds the other tail, because the test is two-tailed.
• The result is about 0.33, well above 0.05, which agrees with the decision to fail to reject.