1.Easy
A machine is supposed to fill bottles with 100 mL on average. The standard deviation is known to be 5 mL. A sample of 100 bottles has a mean of 100.3 mL. Find the test statistic z for H₀: μ = 100.
μ₀ = 100 mLx̄ = 100.3 mLσ = 5 mLn = 100
z =
Hypothesis tests & confidence intervals worksheet with answers
2.Easy
A z test of H₀: μ = μ₀ against H₁: μ < μ₀ gives the test statistic z = −1.34. Find the p-value p.
z = −1.34H₁: μ < μ₀
p =
3.Medium
The mean of a population is claimed to be 100; the population standard deviation is known to be 12. A random sample of 144 observations has a mean of 99.04. Test H₀: μ = 100 against H₁: μ ≠ 100. Find the p-value p.
μ₀ = 100σ = 12n = 144x̄ = 99.04H₁: μ ≠ μ₀
p =
4.Medium
Two production lines make the same part. Line 1: a sample of 80 parts has mean length 43.5 mm, and the line's known standard deviation is 10 mm. Line 2: a sample of 100 parts has mean 44.8 mm, known standard deviation 12 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
n₁ = 80x̄₁ = 43.5 mmσ₁ = 10 mmn₂ = 100x̄₂ = 44.8 mmσ₂ = 12 mm
z =
5.Hard
How large must a sample be so that the 99% confidence interval for a mean has a margin of error of at most 1.5? The standard deviation is 20. Find the smallest sample size n.
σ = 20E = 1.599%
n =
6.Hard
A one-sided z test of H₀: μ = 100 against H₁: μ < 100 is done at α = 0.01 with a sample of 36; the standard deviation is known to be 15. In reality the mean is 93.75. Find the probability of a Type II error β in percent.
μ₀ = 100μ = 93.75σ = 15n = 36α = 0.01
β = %
Answer key
- 1. z = 0.6
- 2. p = 0.09012
- 3. p = 0.3371
- 4. z = −0.7926
- 5. n = 1180
- 6. β = 43.09 %
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