Question

4

We draw a random sample of size 40 from a population with standard deviation 2.5.

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a

If the sample mean is 27, what is a 95% confidence interval for the population mean?

b

If the sample mean is 27, what is a 99% confidence interval for the population mean?

c

If the sample mean is 27, what is a 90% confidence interval for the population mean?

d

If the sample mean is 27 and the sample size is 87, what is a 95% confidence interval for the population mean?

Answer 1

x̅ = 27, σ = 2.5, n = 40

a)

95% Confidence interval :

At α = 0.05 two tailed critical value, z_c = ABS(NORM.S.INV(0.05/2)) = 1.960

Lower Bound = x̅ - z_c*σ/√n = 27 - 1.96 * 2.5/√40 = 26.2253

Upper Bound = x̅ + z_c*σ/√n = 27 + 1.96 * 2.5/√40 = 27.7747

b)

99% Confidence interval :

At α = 0.01 two tailed critical value, z_c = ABS(NORM.S.INV(0.01/2)) = 2.576

Lower Bound = x̅ - z_c*σ/√n = 27 - 2.576 * 2.5/√40 = 25.9818

Upper Bound = x̅ + z_c*σ/√n = 27 + 2.576 * 2.5/√40 = 28.0182

c)

90% Confidence interval :

At α = 0.1 two tailed critical value, z_c = ABS(NORM.S.INV(0.1/2)) = 1.645

Lower Bound = x̅ - z_c*σ/√n = 27 - 1.645 * 2.5/√40 = 26.3498

Upper Bound = x̅ + z_c*σ/√n = 27 + 1.645 * 2.5/√40 = 27.6502

d)

x̅ = 27, σ = 2.5, n = 87

95% Confidence interval :

At α = 0.05 two tailed critical value, z_c = ABS(NORM.S.INV(0.05/2)) = 1.960

Lower Bound = x̅ - z_c*σ/√n = 27 - 1.96 * 2.5/√87 = 26.4747

Upper Bound = x̅ + z_c*σ/√n = 27 + 1.96 * 2.5/√87 = 27.5253