A machine is set to fill packs with a mean of 500 g; you suspect it underfills. The standard deviation is known to be 8 g. You will weigh a sample of 64 packs and test H₀: μ = 500 against H₁: μ < 500 at α = 0.05. Below which sample mean x̄c will you reject H₀?
A z test of H₀: μ = μ₀ against H₁: μ > μ₀ gives the test statistic z = 2.05. Find the p-value p.
Two teaching methods are compared. Group 1: 5 students, mean score 35.5, standard deviation 9. Group 2: 7 students, mean score 38.8, 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).
A sample of 64 students has a mean score of 202.9 and a sample standard deviation of 8. Find the test statistic t for H₀: μ = 200.
A one-sided z test of H₀: μ = 100 against H₁: μ < 100 is done at α = 0.01 with a sample of 25; the standard deviation is known to be 2.5. In reality the mean is 98.9. Find the power 1 − β of the test in percent.
How large must a sample be so that the 90% confidence interval for a mean has a margin of error of at most 1.5? The standard deviation is 5. Find the smallest sample size n.
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