Hypothesis tests & confidence intervals worksheet with answers

A company claims that a proportion of 0.8 of its customers are satisfied. In a random survey of 400 customers, 304 say they are satisfied. Find the test statistic z for H₀: p = 0.8.

A supplier claims that the mean breaking strength of a cable is 50 kN; you suspect it is higher. The standard deviation is known to be 4 kN. You will take a sample of 36 cables and test H₀: μ = 50 kN against H₁: μ > 50 kN at α = 0.10. Above which sample mean x̄_c will you reject H₀?

12 employees take a typing test before and after a training course. For each person the difference d = after − before (words per minute) is recorded. The mean difference is 2.52 and the standard deviation of the differences is 2.3. Find the paired test statistic t for H₀: μ_d = 0.

Two production lines make the same part. Line 1: a sample of 60 parts has mean length 131 mm, and the line's known standard deviation is 16 mm. Line 2: a sample of 100 parts has mean 133.6 mm, known standard deviation 11 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).

A sample of 36 measurements has mean 96.3. The population standard deviation is 8. Find the lower limit L of the 90% confidence interval for the mean.

A one-sided z test of H₀: μ = 500 against H₁: μ > 500 is done at α = 0.05 with a sample of 64; the standard deviation is known to be 16. In reality the mean is 505.4. Find the probability of a Type II error β in percent.