A company receives shipments of a component used in the manufacture of a high-end acoustic speaker system. When the components arrive, the company selects a random sample from the shipment and subjects the selected components to a rigorous set of tests to determine if the components in the shipments conform to their specifications. From a recent large shipment, a random sample of 250 of the components was tested, and 24 units failed one or more of the tests.

a) What is the point estimate of the proportion of components in the shipment that fail to meet the company's specifications?
b) What is the standard error of the estimated proportion?
c) At the 98% level of confidence, what is the margin of error in this estimate?
d) What is the 95% confidence interval estimate for the true proportion of components, p, that fail to meet the specifications?
e) If the company wanted to test the null and alternative hypotheses: H_0: p = 0.10 against H_a: p notequalto 0.10 at the alpha = 0.05 level of significance, what conclusion would they draw?

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MathPhys MathPhys · Answer 1 · 2020-04-15T01:49:11+02:00

Step-by-step explanation:

a) 24 / 250 = 0.096

b) Standard error for a proportion is:

σ = √(pq/n)

σ = √(0.096 × 0.904 / 250)

σ = 0.0186

c) At 98% confidence, the critical value is 2.326. The margin of error is therefore:

2.326 × 0.0186 = 0.0433

d) At 95% confidence, the critical value is 1.960. The margin of error is therefore:

1.960 × 0.0186 = 0.0365

So the confidence interval is:

(0.0960 − 0.0365, 0.0960 + 0.0365)

(0.0595, 0.1325)

e) 0.10 is within the 95% confidence interval, so the null hypothesis would not be rejected.