Answer:
The probability is [tex]P(|\^ p - p | < 0.03) = 0.87224[/tex]
Step-by-step explanation:
From the question we are told that
The population proportion is p = 0.55
The sample size is n = 639
Generally the standard deviation of this sampling distribution is mathematically represented as
[tex]\sigma = \sqrt{\frac{p(1 -p)}{n} }[/tex]
=> [tex]\sigma = \sqrt{\frac{0.55 ( 1 -0.55)}{639 } }[/tex]
=> [tex]\sigma = 0.0197[/tex]
Generally the probability that the proportion of persons with a retirement account will differ from the population proportion by less than 3% is mathematically evaluated as
[tex]P(|\^ p - p | < 0.03) = P( \frac{|\^ p - p |}{\sigma } < \frac{0.03}{0.0197} )[/tex]
[tex]\frac{|\^ p-p|}{\sigma } = |Z| (The \ standardized \ value\ of \ |\^ p - p|)[/tex]
[tex]P(|\^ p - p | < 0.03) = P( {|Z| < 1.523 )[/tex]
=> [tex]P(|\^ p - p | < 0.03) = P( -1.523 \le Z \le 1.523 )[/tex]
=> [tex]P(|\^ p - p | < 0.03) = P( Z \le 1.523) - P( Z \le -1.523 )[/tex]
From the z table the area under the normal curve to the left corresponding to -1.523 is
[tex]P(Z \le -1.523 ) = 0.063879[/tex]
From the z table the area under the normal curve to the left corresponding to 1.523 is
[tex]P(Z \le 1.523 ) = 0.93612[/tex]
[tex]P(|\^ p - p | < 0.03) = 0.93612 - 0.063879[/tex]
[tex]P(|\^ p - p | < 0.03) = 0.87224[/tex]
