Question

A programmer plans to develop a new software system. In planning for the operating system that he will​ use, he needs to estimate the percentage of computers that use a new operating system. How many computers must be surveyed in order to be

9595​%

confident that his estimate is in error by no more than

fivefive

percentage

points question mark s?

Complete parts​ (a) through​ (c) below.

​a) Assume that nothing is known about the percentage of computers with new operating systems.

nequals=nothing

​(Round up to the nearest​ integer.)

​b) Assume that a recent survey suggests that about

8989​%

of computers use a new operating system.

nequals=nothing

​(Round up to the nearest​ integer.)

​c) Does the additional survey information from part​ (b) have much of an effect on the sample size that is​ required?

A.

​Yes, using the additional survey information from part​ (b) dramatically increases the sample size.

B.

​No, using the additional survey information from part​ (b) does not change the sample size.

C.

​Yes, using the additional survey information from part​ (b) dramatically reduces the sample size.

D.

​No, using the additional survey information from part​ (b) only slightly increases the sample size.

Answer 1

a)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.05

The provided estimate of proportion p is, p = 0.5
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.5*(1 - 0.5)*(1.96/0.05)^2
n = 384.16

Therefore, the sample size needed to satisfy the condition n >= 384.16 and it must be an integer number, we conclude that the minimum required sample size is n = 385
Ans : Sample size, n = 385 or 384

b)

The following information is provided,
Significance Level, α = 0.05, Margin of Error, E = 0.05

The provided estimate of proportion p is, p = 0.89
The critical value for significance level, α = 0.05 is 1.96.

The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:
n >= p*(1-p)*(zc/E)^2
n = 0.89*(1 - 0.89)*(1.96/0.05)^2
n = 150.44

Therefore, the sample size needed to satisfy the condition n >= 150.44 and it must be an integer number, we conclude that the minimum required sample size is n = 151
Ans : Sample size, n = 151 or 150

c)

C.

​Yes, using the additional survey information from part​ (b) dramatically reduces the sample size.

D.