Question

Yobel is a supply chain management company in Peru. They manufacture on-site products for large international companies such as Revlon, Kodak, Ivory, etc. These companies have specific quality criteria that they expect to be maintained even if the product is not manufactured or sold in the United States. Suppose you're put in charge of quality control for Revlon lipstick. The specified "failure rate" or proportion of flawed lipsticks, as specified by Revlon is 0.05. This sort of relationship is normally the case when a company outsources some of its production.

a) You sample 100 lipsticks and find that 7 of them are flawed. What is the proportion?

b) Calculate a 95% confidence interval for the failure rate of lipsticks.

c) Test to see if your factory has a higher failure rate than the company specified level.

d) Suppose that you have a specified margin of error of 0.01 and a required confidence level of 99%. What is the minimum number of lipsticks that you need to test to be this accurate?

e) What if instead your margin of error was 0.05. Now how many lipsticks do you need to test?

Answer 1

(a) = 7 / 100 = 0.07

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(b) 95% CI. Z critical = 1.96

The Lower Limit = 0.07 - 0.05 = 0.02

The Upper Limit = 0.07 + 0.05 = 0.12

The Confidence Interval is (0.02 , 0.12)

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(c) The Hypothesis:

H0: p = 0.05

Ha: p > 0.05

This is a Right tailed Test.

The Test Statistic:

the critical Value is 1.96.

Since Z test is < 1.96, We fail to Reject H0.

There isn't sufficient evidence at the 95% level to conclude that the failure rate of our company is higher than the company specified level.

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(d) The ME = Zcritical * SQRT(p * q/n).

Squaring and solving, n = (Zc/ME)² * p * q

Here ME = 0.01, p = 0.07, q = 0.93

The Zc at = 0.01 is 2.576

Therefore n = (2.576/0.01)² * 0.07 * 0.93 = 4319.89

Therefore n = 4320 (Rounding to the nearest Integer)

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(e) The Zc at = 0.05 is 1.96

Therefore n = (1.96/0.01)² * 0.07 * 0.93 = 2500.88

Therefore n = 2501 (Rounding to the nearest Integer)

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