Question

2. A semiconductor factory initially estimated that 1% of its products have a faulty packaging and want to show that actual fault rate does not exceeds this initial estimate. A random sample of 1200 products yielded 19 defected ones. Test the initial estimation with this.

a. Calculate the test statistic Z.

b. Test the company’s initial estimation (H0) with 90% confidence level. Use one-tail test. Calculation process of the test must be described.

c. Referring Problem 2, what is the probability that the above test fails to detect the H0 is false when actual faulty packaging probability is 2%?

Answer 1

Question 2

H0 : p = 0.01

Ha : p > 0.01

(a) Here np = 1200 * 0.01 = 12 > 5 so we would approximate it with normal distribution

Standard error = sqrt [p(1-p)/n] = sqrt [0.01 * 0.99/1200] = 0.002872

p^ = 19/1200 = 0.015833

Z = (0.015833 - 0.01)/0.002872 = 2.0309

(b) 90% confidence interval = p^ +- Zcritical sep = 0.01 +- 1.645 * 0.002872 = (0.005275, 0.014725)

so here the sample proporton is not inside the confidence interval

so we would reject the null hypothesis and conclude that proportion of defective is greater than 1% .

(c) actual faulty packaging probability is 2% and we would fail to reject the null hypothesis when p^ < 0.014725

standard error = sqrt (0.02 * 0.98/1200) = 0.004041

P(p^ < 0.015833; 0.02; 0.004041)

z = (0.014725 - 0.02)/0.004041 = -1.3054

P(p^ < 0.015833; 0.02; 0.004041) = P(Z < -1.3054) = 0.096

the probability that the above test fails to detect the H0 is false is 0.096.