A supplier claims that the mean breaking strength of a cable is 120 kN; you suspect it is higher. The standard deviation is known to be 10 kN. You will take a sample of 36 cables and test H₀: μ = 120 kN against H₁: μ > 120 kN at α = 0.05. Above 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 10 mL. A sample of 64 bottles has a mean of 500.9 mL. Find the test statistic z for H₀: μ = 500.
Two teaching methods are compared. Group 1: 8 students, mean score 97.5, standard deviation 6.5. Group 2: 23 students, mean score 94.7, standard deviation 5.5. 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).
Two production lines make the same part. Line 1: a sample of 100 parts has mean length 41 mm, and the line's known standard deviation is 17 mm. Line 2: a sample of 80 parts has mean 39.2 mm, known standard deviation 14 mm. Find the test statistic z for H₀: μ₁ = μ₂, taking the difference as x̄₁ − x̄₂ (line 1 minus line 2).
A one-sided z test at α = 0.05 must detect a shift of the mean of 2.5 units with a power of 80%. The standard deviation is 9. Find the minimum sample size n.
In a trial, 113 of 250 patients on treatment A recover, and 72 of 120 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
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