Answer:
The probability is [tex]P(0.35 <X < 0.55) = 0.9921772[/tex]
Step-by-step explanation:
From the question we are told that
The sample size is n = 175
The population proportion is p = 0.45
Generally the mean of the sampling distribution is [tex]\mu_{x} = 0.45[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma_{x} = \sqrt{\frac{p (1-p)}{n} }[/tex]
=> [tex]\sigma_{x} = \sqrt{\frac{0.45 (1-0.45)}{175} }[/tex]
=> [tex]\sigma_{x} = 0.0376[/tex]
Generally the probability of that the sample proportion of orange candies will be between 0.35 and 0.55 is
[tex]P(0.35 < X < 0.55) = P( \frac{0.35 - 0.45}{0.0376} < \frac{X -\mu_{x}}{\sigma_{x}} < \frac{0.55 - 0.45}{0.0376} )[/tex]
=> [tex]P(0.35 <X < 0.55) = P( -2.696 < \frac{X -\mu_{x}}{\sigma_{x}} < 2.6595 )[/tex]
Generally [tex]\frac{X - \mu_{x}}{\sigma_{x}} = Z (The \ standardized \ value \ of X)[/tex]
So
[tex]P(0.35 <X < 0.55) = P( -2.6596 < Z< 2.6595 )[/tex]
=> [tex]P(0.35 <X < 0.55) =P( Z< 2.6595 ) - P( Z < -2.6596 )[/tex]
From the z-table
[tex]P( Z< 2.6595 ) = 0.99609[/tex]
and
[tex]P( Z< - 2.6595 ) = 0.0039128[/tex]
So
[tex]P(0.35 <X < 0.55) =0.99609 - 0.0039128[/tex]
=> [tex]P(0.35 <X < 0.55) = 0.9921772[/tex]
