Answer:
The value is [tex]P(X < 0.50 ) = 0.43133[/tex]
Step-by-step explanation
From the question we are told that
The population proportion is p = 0.51
The sample size is n = 75
Generally given that the sample size is large enough (i.e n > 30) the mean of this sampling distribution is mathematically represented as
[tex]\mu_x = p = 0.5 1[/tex]
Generally the standard deviation of this sample distribution is mathematically represented as
[tex]\sigma = \sqrt{\frac{p(1- p)}{ n} }[/tex]
=> [tex]\sigma = \sqrt{\frac{0.51 (1- 0.51 )}{75} }[/tex]
=> [tex]\sigma = 0.058[/tex]
Generally the probability that in a random sample of 75 voters, fewer than 50% of the sample will vote for Candidate A is mathematically represented as
[tex]P(X < 0.50 ) = P( \frac{ X - \mu_x }{\sigma } < \frac{ 0.50 - 0.51 }{0.0578 } )[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
=> [tex]P(X < 0.50 ) = P( Z < -0.1730 )[/tex]
From the z table the area under the normal curve to the left corresponding to -0.1720 is
[tex]P( Z < -0.1730 ) = 0.43133[/tex]
So
[tex]P(X < 0.50 ) = 0.43133[/tex]
