The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material. The sample mean and standard deviation for alloy 1 was x̅₁ = 420°F and s₁ = 4°F, and for alloy 2, they were x̅₂ = 426°F and s₂ = 3°F.

(a) Do the sample data support the claim that both alloys have the same melting point? Use α = 0.05 and assume that both populations are normally distributed and have the same standard deviation. Find the P-value for the test.

(b) Suppose that the true mean difference in melting points is 3°F. How large a sample would be required to detect this difference using an α = 0.05 level test with probability at least 0.9? Use σ₁ = σ2 = 4 as an initial estimate of the common standard deviation.