Question

Two different types of alloy, A and B, have been used to manufacture experimental specimens of a small tension link to be used in a certain engineering application. The ultimate strength (ksi) of each specimen was determined, and the results are summarized in the accompanying frequency distribution.

	\(A\)	\({\bf{B}}\)
\(26 - < 30\)	6	4
\(30 - < 34\)	12	9
\(34 - < 38\)	15	19
\(38 - < 42\)	7	10
	\(m = 40\)	\(m = 42\)

Compute a 95 % CI for the difference between the true proportions of all specimens of alloys A and B that have an ultimate strength of at least\(34ksi\).

Short answer

The 95% confidence interval is from -0.35 to 0.07.

Step-by-step solution

Step 1: To compute a 95 % CI for the difference between the true proportions of all specimens of alloys a and B

A confidence interval for \({p_1} - {p_2}\)with confidence level of approximately \(100(1 - \alpha )\% \)is

\({\hat p_1} - {\hat p_2} \pm {z_{\alpha /2}} \times \sqrt {\frac{{{{\hat p}_1}{{\hat q}_1}}}{m} + \frac{{{{\hat p}_2}{{\hat q}_2}}}{n}} \)

The confidence interval can be used when\(m \times {\hat p_1},m \times {\hat q_1},n \times {\hat p_1}\), and\(n \times {\hat q_1}\)are at least 10.

The proportion of specimens of alloys A that have an ultimate strength of at least \(34{\rm{ksi}}\) is

\({\hat p_1} = \frac{{15 + 7}}{{40}} = 0.55\)

The proportion of specimens of alloys B that have an ultimate strength of at least \(34{\rm{ksi}}\) is

\({\hat p_1} = \frac{{19 + 10}}{{40}} = 0.69.\)

Finding confidence interval

For the 95 % confidence interval \(\alpha = 0.05\)and\({z_{\alpha /2}} = {z_{0.025}} = 1.96\). Thus, the \(95\% \)confidence interval is

\(\begin{array}{c}{{\hat p}_1} - {{\hat p}_2} \pm {z_{\alpha /2}} \times \sqrt {\frac{{{{\hat p}_1}{{\hat q}_1}}}{m} + \frac{{{{\hat p}_2}{{\hat q}_2}}}{n}} \\ = 0.55 - 0.69 \pm 1.96 \times \sqrt {\frac{{0.55 \times 0.45}}{{40}} + \frac{{0.69 \times 0.31}}{{42}}} \\ = - 0.14 \pm 1.96 \times 0.106\\ = ( - 0.35,0.07).\end{array}\)

The 95% confidence interval for the difference between the two proportions is from -0.35 to 0.07