A machine is set to fill packs with a mean of 200 g; you suspect it underfills. The standard deviation is known to be 15 g. You will weigh a sample of 25 packs and test H₀: μ = 200 against H₁: μ < 200 at α = 0.10. Below which sample mean x̄c will you reject H₀?
A company claims that a proportion of 0.3 of its customers are satisfied. In a random survey of 120 customers, 26 say they are satisfied. Find the test statistic z for H₀: p = 0.3.
Two production lines make the same part. Line 1: a sample of 60 parts has mean length 72 mm, and the line's known standard deviation is 13 mm. Line 2: a sample of 30 parts has mean 74.3 mm, known standard deviation 9 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: 6 students, mean score 24.5, standard deviation 6. Group 2: 8 students, mean score 22.2, standard deviation 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 25 measurements has mean 106. The population standard deviation is 10. Find the upper limit U of the 95% confidence interval for the mean.
A one-sided z test at α = 0.05 must detect a shift of the mean of 2 units with a power of 95%. The standard deviation is 7. Find the minimum sample size n.
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