Hypothesis tests & confidence intervals worksheet with answers

A machine is supposed to fill bottles with 50 mL on average. The standard deviation is known to be 5 mL. A sample of 100 bottles has a mean of 51.45 mL. Find the test statistic z for H₀: μ = 50.

A machine is set to fill packs with a mean of 75 g; you suspect it underfills. The standard deviation is known to be 8 g. You will weigh a sample of 16 packs and test H₀: μ = 75 against H₁: μ < 75 at α = 0.01. Below which sample mean x̄_c will you reject H₀?

9 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 12.41 and the standard deviation of the differences is 9.8. Find the paired test statistic t for H₀: μ_d = 0.

Two teaching methods are compared. Group 1: 19 students, mean score 85, standard deviation 8.5. Group 2: 12 students, mean score 96.2, standard deviation 11. Assume equal population variances and use the pooled two-sample t test. Find the test statistic t for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (group 1 minus group 2).

A sample of 49 measurements has mean 102.5. The population standard deviation is 8. Find the upper limit U of the 90% confidence interval for the mean.

How large must a sample be so that the 90% confidence interval for a mean has a margin of error of at most 0.5? The standard deviation is 5. Find the smallest sample size n.