Mathematics: Given a population where the probability of success is p= 0.40, calculate the probabilities below if a sample of is taken. Calculate the probability the proportion of successes in the sample will be . What is the probability the proportion of successes in the sample will be ?

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Given a population where the probability of success

alvalynnw

07.02.2022

Complete Question

Given a population where the probability of success is p= 0.40 calculate the probabilities below if a sample of 300 is taken.

A. Calculate the probability the proportion of successes in the sample will be less than 0.42 (round 4 decimals)

B. What is the probability that the proportion of successes in the sample will be greater than 0.44 (round 4 decimals)

P(X < 0.42) = 0.76028

P(X 0.44) = 0.078622

Step-by-step explanation:

From the question we are told that

The probability of success is p = 0.40

The sample size is n = 300

Generally given that the sample size is large enough n > 30 then the mean for this sampling distribution is

$\mu_{x} = p = 0.40$

Generally the standard deviation is mathematically represented as

$\sigma = \sqrt{ \frac{p (1 - p )}{n} }$

=> $\sigma = \sqrt{ \frac{0.40 (1 - 0.40 )}{ 300} }$

=> $\sigma = 0.02828$

Considering question A

Generally the probability the proportion of successes in the sample will be less than 0.42 is mathematically represented as

$P(X < 0.42) = P(\frac{X - \mu }{\sigma } < \frac{0.42 - 0.40 }{ 0.02828} )$

$\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )$

=> P(X < 0.42) = P(Z < 0.7072 )

From the z table

The area under the normal curve to the left corresponding to 0.7072 is

P(Z < 0.7072 ) = 0.76028

=> P(X < 0.42) = 0.76028

Considering question B

Generally the probability the proportion of successes in the sample will be less than 0.44 is mathematically represented as

$P(X 0.44) = P(\frac{X - \mu }{\sigma } \frac{0.44 - 0.40 }{ 0.02828} )$

$\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )$

=> P(X 0.44) = P(Z 1.4144 )

From the z table

The area under the normal curve to the left corresponding to 1.4144 is

P(Z 1.4144 ) = 0.078622

=> P(X 0.44) = 0.078622

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