The one-sample t test for a mean

Three quick questions from earlier lessons.

H₀: μ = 50. A sample of 16 has x̄ = 53 and s = 8. How many standard errors is 53 above 50?

The one-sample t test.

t = \dfrac{\bar{x} - \mu_0}{s/\sqrt{n}}

with

n - 1

degrees of freedom. For a two-sided Hₐ, the P-value counts both tails (double the one-tail area).

H₀: μ = 100, Hₐ: μ ≠ 100; $n = 25$ , $\bar{x} = 104$ , $s = 10$ .

SE = 2

10/5.

t = 2

, df

= 24

4/2.

P-value

\approx 2 \times 0.028 = 0.057

Both tails.

Your turn: one-tail area 0.04, two-sided test. The P-value.

0.08

2 × 0.04.

Test H₀: μ = 39. A random sample of 9 has x̄ = 43 and s = 9. Find the t statistic.
Test H₀: μ = 24. A random sample of 64 has x̄ = 29 and s = 18. Find the t statistic.

For a test statistic, the area in one tail beyond it is 0.04. Find the P-value of the two-sided test.

A t test gives P-value 0.20. What can you conclude at α = 0.05?
1. Fail to reject H₀: not enough evidence for Hₐ
2. H₀ is true
3. Reject H₀

Three questions to check you have got it.

Test H₀: μ = 39. A random sample of 9 has x̄ = 43 and s = 9. Find the t statistic.
Test H₀: μ = 24. A random sample of 64 has x̄ = 29 and s = 18. Find the t statistic.
Test H₀: μ = 71. A random sample of 9 has x̄ = 76 and s = 8. Find the t statistic.

t test for μ

$t = (\bar{x} - \mu_0)/(s/\sqrt{n})$ , df $= n - 1$ ; two-sided: double the tail.

Watch out: failing to reject is not proving H₀.

End of the lesson