A machine is set to fill packs with a mean of 50 g; you suspect it underfills. The standard deviation is known to be 12 g. You will weigh a sample of 16 packs and test H₀: μ = 50 against H₁: μ < 50 at α = 0.05. Below which sample mean x̄c will you reject H₀?
A company claims that a proportion of 0.5 of its customers are satisfied. In a random survey of 120 customers, 55 say they are satisfied. Find the test statistic z for H₀: p = 0.5.
A sample of 36 students has a mean score of 56 and a sample standard deviation of 15. Find the test statistic t for H₀: μ = 50.
Two teaching methods are compared. Group 1: 6 students, mean score 34, standard deviation 9.5. Group 2: 6 students, mean score 40.3, 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).
In a trial, 60 of 150 patients on treatment A recover, and 72 of 150 patients on treatment B. Using the pooled proportion, find the test statistic z for H₀: p₁ = p₂.
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 10. Find the smallest sample size n.
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