A machine is set to fill packs with a mean of 75 g; you suspect it underfills. The standard deviation is known to be 5 g. You will weigh a sample of 16 packs and test H₀: μ = 75 against H₁: μ < 75 at α = 0.05. Below which sample mean x̄c will you reject H₀?
A machine is supposed to fill bottles with 500 mL on average. The standard deviation is known to be 4 mL. A sample of 49 bottles has a mean of 500.7 mL. Find the test statistic z for H₀: μ = 500.
Two production lines make the same part. Line 1: a sample of 30 parts has mean length 97 mm, and the line's known standard deviation is 8 mm. Line 2: a sample of 50 parts has mean 92.1 mm, known standard deviation 5 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
Two teaching methods are compared. Group 1: 18 students, mean score 115, standard deviation 5.5. Group 2: 13 students, mean score 118.7, standard deviation 2. 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 one-sided z test at α = 0.05 must detect a shift of the mean of 1 units with a power of 95%. The standard deviation is 5. Find the minimum sample size n.
How large must a sample be so that the 90% confidence interval for a mean has a margin of error of at most 2? The standard deviation is 5. Find the smallest sample size n.
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