Answer to Question #253335 in Statistics and Probability for bao

Question

The article “Evaluating Tunnel Kiln Performance” (Amer. Ceramic Soc. Bull., Aug. 1997: 59–63) gave the following summary information for fracture strengths (MPa) of n=169

n=169 ceramic bars fired in a particular kiln: x

¯

=89.10,σ=3.73

x¯=89.10,σ=3.73

a. Calculate a (two-sided) confidence interval for true average fracture strength using a confidence level of 95%

95%.

b. Suppose the investigators had believed a priori that the population standard deviation was about 4MPa

4MPa. Based on this supposition, how large a sample would have been required to estimate μ

μ to within .5 MPa with 95%

95% confidence?

Answer 1

a. α = 0.05

$df = n-1 = 168 \\ t_c = 1.974 \\ CI = \bar{x} ± t_c \times \frac{s}{\sqrt{n}} \\ CI = 89.1 ± 1.974 \times \frac{3.73}{\sqrt{169}} \\ CI = 89.1 ± 0.566 \\ CI = (88.534, 89.666)$

b.

$Error = Z \times \frac{\sigma }{ \sqrt{n}} \\ n = (\frac{Z \times \sigma}{error})^2 \\ for \; α=0.05 \\ Z = 1.96 \\ n = (\frac{1.96 \times 4}{0.5})^2 \\ n = 245.86 \\ n = 246$