What is the highest rate of non defective CD players that the manufacturer could claim under the following condition?

Question

A manufacturer finds that in a random sample of 100 of its CD players, 96% have no defects. The manufacturer wishes to make a claim about the percentage of non defective CD players and is prepared to exaggerate. His claim would not be rejected at the 0.05 significance level if this sample data were used. Assume that a left-tailed hypothesis test would be performed.

Answer 1

The manufacturer wants to claim the smallest proportion of defects that will not be thrown into doubt by the specified sample. Let's denote that proportion by x, and denote the population proportion of defects by p.

Then the null hypothesis is H₀: p = x

and the (one-sided) alternative hypothesis is H_A: p > x

We have a specified sample, and we want this sample not to cause us to reject the null hypothesis at the 95% significance level for a one-tailed test.

Thus we want the probability to be 95% that the sample is within the confidence interval.

Specifically, we want the 95% confidence interval to be (0, x + 1.64 SD) where SE is the standard deviation for the sampling distribution of a proportion with a hypothesized probability of x. SD is √(x(1-x)/n

Now, we have n = 100, so SD = √(x(1-x)/100

We further have the sample defect proportion to be .04 (4 out of 100).

Since we want this sample not to cause us to reject the null hypothesis,

this sample defect can be the highest value of the confidence interval, but no higher.

Thus, .04 = x + 1.64 √(x(1-x)/100

This needs a bit of algebra to solve for x.

Subtract x from both sides to get

.04 - x = 1.64 √(x(1-x)/100

Square both sides:

.0016 - .08x + x² = 2.6896 x(1-x) / 100

.0016 - .08x + x² = .0026896 x(1 - x)

0016 - .08x + x² = .0026896 x - .0026896 x²

Combining terms give:

1.0026896 x² - .0826896 x + .0016 = 0

Use the quadratic formula, or some other solver, to solve for x in the above equation.

x ≈ .031 for the smaller value

So, the manufacturer can claim the defect rate for his CD players is only 3.1%, and the sample will not cause this hypothesis to be rejected at the .05 significance level.