98% confidence interval for the mean rate = [4.1437 , 4.4983]
We are given the interest rates (annual percentage rates) for a 30-year fixed rate mortgage from a sample of lenders in Macon, Georgia for one day ;
4.751, 4.373, 4.177, 4.676, 4.425, 4.228, 4.125, 4.251, 3.951, 4.192, 4.291, 4.414
Now, Firstly we will find the Mean of the above data, Xbar ;
Mean, Xbar = ∑ x ÷ n =
4.751 + 4.373 + 4.177 + 4.676 + 4.425 + 4.228 + 4.125 + 4.251 + 3.951 + 4.192 + 4.291 + 4.414 ÷ 12 = 4.321
Standard deviation, s = √ ∑(x - x bar)²÷ n-1 = 0.226.
Now, the pivotal quantity for the 98% confidence interval for the mean rate is;
P.Q = x bar - ц ÷ n - 1 ≈ tn - 1
where, Xbar = sample mean
s = sample standard deviation
n = sample size = 12
So, 98% confidence interval for the mean rate, μ is ;
P(-2.718 < t₁₁ <2.718) = 0.98
P(-2.718 <Xbar - μ σ√ⁿ < 2.718) = 0.98
P(Xbar - 2.718 * ₈÷ √ⁿ < μ Xbar + 2.718 * ₈÷ √ⁿ ) = 0.98
98% confidence interval for μ = (Xbar - 2.718 * ₈÷ √ⁿ < μ Xbar + 2.718 * ₈÷ √ⁿ ) = 0.98
[4.321 - 2.718 * 0.226 ÷ √₁₂ , 4.321 + 2.718 * 0.226 ÷ √₁₂
= [4.1437 , 4.4983]
Therefore, 98% confidence interval for the mean rate = [4.1437 , 4.4983]
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