Question 1088405
85*0.059=5 defects
Use binomial
probability 5 or more defects in 85 given that the rate is 3%: That is nCx (probability of  failure^x)(probability of success^(n-x))
85C5(.03^5)(.97^80)=0.0697 for 5
there will be non=zero values for 6, 7,, 8...
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use 1 sample proportion with np at the borderline of acceptable (np=3)
Critical value is z>2,33 for 1 way test.
z=(.059-0.03)/sqrt (p*(1-p))/n
sqrt term is .03*.97/85=0.0185
z=0.029/0.0185=1.56
Not significant at the 0.01 level.
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For the second one, can do a confidence interval as well as a z-test
Will do the z-test
one sample proportion same as last part
critical value z>1,645 for a 1-way test
z=(0.50-0.48)/sqrt (0.5*0.5/1068)
=0.02/0.0153=1.31
Cannot exclude 50% as a plausible percentage based on these data.
quick check: 1/sqrt (sample size) is approximate error for 2 sided test, and that is +/-0.03, so that 50% is plausible.