Question

Two processes are used to produce forgings used in an aircraft wing assembly. Of 200 forgings selected from process 1, 10 do not conform to the strength specifications, whereas of 300 forgings selected from process 2, 20 are nonconforming. a) Esetimate the fraction nonconforming for each process. b) Test the hypothesis that the two process have identical fractions nonconforming. Use alpha =0.05. c) Construct a 90% confidence interval on the difference in fraction nonconforming between the two processes.

Answer 1

a.

`\bar p_1=0.05\\\bar p_2=0.067`

b-Check illustration below

c.(-0.0517,0.0177

Explanation:

a.let
`p_1 \& p_2` denote processes 1 & 2.

For
`p_1`: T1=10,n1=200

For
`p_2`:T2=20,n2=300

Therefore

`\bar p_1=(t_1)/(N_1)=(10)/(200)=0.05\\\bar p_2=(t_2)/(N_2)=(20)/(300)=0.067`

b. To test for hypothesis:-

i.

`H_0:p_1=p_2\\H_A=p_1\\eq p_2\\\alpha=0.05`

ii.For a two sample Proportion test

`Z=(\bar p_1-\bar p_2)/(\sqrt(\bar p(1-\bar p)((1)/(n_1)+(1)/(n_2)))\\`

iii. for
`(\alpha)/(2)=(-1.96,+1.96)` (0.5 alpha IS 0.025),

reject
`H_o` if
`|Z|>1.96`

iv. Do not reject
`H_o`. The noncomforting proportions are not significantly different as calculated below:

`z=\frac{0.050-0.067}{\sqrt {(0.06*0.94)* (1)/(500)}}`

z=-0.78

c.
`(1-\alpha).100\%` for the p1-p2 is given as:

`(\bar p_1-\bar p_2)\pm Z_0_._5_\alpha * \sqrt ( \bar p_1(1-\bar p_1))/(n_1)+(\bar p_2(1-\bar p_2))/(n_2)\\\\=(0.05-0.067)\pm 1.645 * \sqrt \ (0.05+0.95)/(200)+(0.067+0.933)/(300)\\`

=(-0.0517,+0.0177)

*CI contains o, which implies that proportions are NOT significantly different.