Answered: Why is the formula of the standard…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Why is the formula of the standard error of D hat is the square root of 1/4 times (0.156 squared + 0.0219 squared)? Where did we get the 1/4?

(i) First compute p
S
=
=
=
104
400
= 0.26. So D=
0.11 +0.26
2
(ii) The standard error of P = 0.0156 is obtained from part (b). The standard error of p, is
py (1 - py) (0.26)(0.74)
400
= 0.0219. So the standard error of Ô is
n
√
= 0.185.
(0.0156² +0.0219²) = 0.0134.
=

Transcribed Image Text:A large company produces an equal number of brand-name lightbulbs and generic lightbulbs. The director of quality control sets guidelines that production will be stopped if there is evidence that the proportion of all lightbulbs that are defective is greater than 0.10. The director also believes that the proportion of brand-name lightbulbs that are defective is not equal to the proportion of generic lightbulbs that are defective. Therefore, the director wants to estimate the average of the two proportions. To estimate the proportion of brand-name lightbulbs that are defective, a simple random sample of 400 brand-name lightbulbs is taken and 44 are found to be defective. Let X represent the number of brand-name lightbulbs that are defective in a sample of 400, and let px represent the proportion of all brand-name lightbulbs that are defective. It is reasonable to assume that X is a binomial random variable. The standard error of px is approximately 0.0156. To estimate the proportion of generic lightbulbs that are defective, a simple random sample of 400 generic lightbulbs is taken and 104 are found to be defective. Let Y represent the number of generic lightbulbs that are defective in a sample of 400. It is reasonable to assume that Y is a binomial random variable and the distribution of py is approximately normal, with an approximate standard error of 0.0219. It is also reasonable to assume that X and Y are independent. The parameter of interest for the manager of quality control is D, the average proportion of defective lightbulbs Px + Py for the brand-name and the generic lightbulbs. D is defined as D = 2 (d) Consider D, the point estimate of D. (i) Calculate D using data from the sample of brand-name lightbulbs and the sample of generic lightbulbs. (ii) Calculates, the standard error of D.

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