Question

A door delivery florist wishes to estimate the proportion of people in his city that will purchase his flowers. Suppose the true proportion is 0.05. If 408 are sampled, what is the probability that the sample proportion will differ from the population proportion by greater than 0.03?

Answer 1

Answer:

 The probability is  $P( | \^ p - p| > 0.03) =0.9945$

Step-by-step explanation:

From the question we are told that

   The population proportion is  $p = 0.05$

    The sample size is  n =  408

Generally the standard deviation of the sampling distribution is mathematically represented as

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

=>    $\sigma = \sqrt{\frac{ 0.05 (1- 0.05)}{408} }$

=>    $\sigma = 0.0108$

Generally the probability that the sample proportion will differ from the population proportion by greater than 0.03 is mathematically represented as

        $P( | \^ p - p| > 0.03) = P( \frac{|\^ p - p | }{\sigma } > \frac{0.03}{0.0108} )$

$\frac{|\^ p -p |}{\sigma }  =  |Z| (The  \ standardized \  value\  of  \ |\^ p -p | )$

So    

       $P( | \^ p - p| > 0.03) = P( |Z|> 2.778 )$

=>   $P( | \^ p - p| > 0.03) = P( Z < 2.778) - P(Z < -2.777)$

From the z table  the area under the normal curve to the left corresponding to  2.778  and  -2.778  is

     $P( Z < 2.778) = 0.99727$

and

     $P( Z < -2.778) = 0.0027347$

So

     $P( | \^ p - p| > 0.03) = 0.99727 - 0.0027347$

=>  $P( | \^ p - p| > 0.03) =0.9945$