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# Probability of Proportions

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A given population proportion is .25. For the given value of n, what is the probability of getting each of the following sample proportions?

a. n = 110 and p̂ ≤ .21

b. n = 33 and p̂ > .24

c. n = 59 and .24 ≤ p̂ < .27

d. n = 80 and p̂ < .30

e. n = 800 and p̂ < .30

https://brainmass.com/statistics/sample-size-determination/probability-proportions-502074

#### Solution Preview

7.22. A given population proportion is .25. For the given value of n, what is the probability of getting each of the following sample proportions?
a. n = 110 and p̂ ≤ .21
Here we can assume that p follows normal distribution with µ = 0.25 and standard deviation σ = = = 0.041286.
We need P (p̂ ≤ .21). Standardizing p using and from standard normal tables, we have
P (p̂ ≤ .21) = = P (Z < -0.96885) = 0.16631
Details
Normal Probabilities

Common Data
Mean 0.25
Standard Deviation 0.041286

Probability for X <=
X Value 0.21
Z Value -0.968851427
P(X<=0.21) 0.166309662

b. n = 33 and p̂ > .24
Here we can assume that p follows normal distribution with µ = 0.25 and standard deviation σ = = = 0.075378.
We need P (p̂ > .24). Standardizing p using and from standard normal tables, we have
P (p̂ > .24) = = P (Z > -0.132665) = 0.5528
Details
Normal Probabilities

Common Data
Mean 0.25
Standard Deviation 0.075378

Probability for X >
X Value 0.24
Z Value -0.132664703
P(X>0.24) 0.5528

c. n = 59 and .24 ≤ p̂ < .27
Here we can assume that p follows normal distribution with µ = 0.25 and standard deviation σ = = = 0.056373.
We need P (.24 ≤ p̂ < .27). Standardizing p using and from standard normal tables, we have
P (.24 ≤ p̂ < .27) =
= P (-0.17739 < Z < 0.35478)
= 0.2090
Details
Normal Probabilities

Common Data
Mean 0.25
Standard Deviation 0.056373
Probability for a Range
From X Value 0.24
To X Value 0.27
Z Value for 0.24 -0.177389885
Z Value for ...

#### Solution Summary

The probabilities of proportions are examined.

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