Question

A telephone exchange operator assumes that 8% of the phone calls are wrong numbers. If the operator is right, what is the probability that the proportion of wrong numbers in a sample of 421 phone calls would differ from the population proportion by greater than 3%? Round your answer to four decimal places.

Answer 1

Answer:

The probability that the sample proportion differ from the population proportion by greater than 3% is 0.0241.

Step-by-step explanation:

Let X = number of phone calls that are wrong numbers.

The proportion of phone calls that are wrong numbers is, p = 0.08.

A sample of n = 421 phone calls is selected to determine the proportion of wrong numbers in this sample.

The random variable X follows a Binomial distribution with parameters n and p.

The probability mass function of a Binomial distribution is:

$P(X=x)={421\choose x}0.08^{x}(1-0.08)^{421-x}$

Now, for the sample proportion to differ from the population proportion by 3% the value of the sample proportion should be:

$\hat p-p=0.03\\\hat p-0.08=0.03\\\hat p=0.11$                            $\hat p-p=-0.03\\\hat p-0.08=-0.03\\\hat p=0.05$

So when the sample proportion is less than 5% or greater than 11% the difference between the sample proportion and population proportion will be greater than 3%.

• If sample proportion is 5% then the value of X is,

        $X=np=421\times 0.05=21.05\approx21$

        Compute the value of P (X ≤ 21) as follows:

       $P(X\leq 21)=\sum\limits^{21}_{x=0}{{421\choose x}0.08^{x}(1-0.08)^{421-x}}=0.0106$

• If the sample proportion is 11% then the value of X is,

        $X=np=421\times 0.11=46.31\approx47$

        Compute the value of P (X ≥ 47) as follows:

       $P(X\geq 47)=\sum\limits^{471}_{x=47}{{421\choose x}0.08^{x}(1-0.08)^{421-x}}=0.0135$

Then the probability that the sample proportion differ from the population proportion by greater than 3% is:

$P(\hat p-p>0.03)=P(X\leq 21)+P(X\geq 47)=0.0106+0.0135=0.0241$

Thus, the probability that the sample proportion differ from the population proportion by greater than 3% is 0.0241.