Answer:
The probability is [tex]P( 3.95< X < 4.05 ) = 0.98758[/tex]
Step-by-step explanation:
From the question we are told that
The population mean is [tex]\mu = 4 \ ounce[/tex]
The standard deviation is [tex]\sigma = 0.22[/tex]
The sample size is n = 121
Generally the standard error of mean is mathematically represented as
[tex]\sigma_{x} = \frac{\sigma}{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{ 0.22}{\sqrt{121} }[/tex]
=> [tex]\sigma_{x} = 0.02[/tex]
The upper limit of the interval within which the sample is consider be found in is
[tex]b = \mu + 0.05[/tex]
=> [tex]b = 4 + 0.05[/tex]
=> [tex]b = 4 .05[/tex]
The lower limit of the interval within which the sample is consider be found in is
[tex]a = \mu - 0.05[/tex]
=> [tex]a= 4 -0.05[/tex]
=> [tex]a = 3.95[/tex]
Generally the probability of obtaining a sample mean within 0.05 ounces of the population mean is mathematically represented as
[tex]P( a < X < b ) = P( \frac{a - \mu }{\sigma } < \frac{a - \mu }{\sigma } < \frac{b - \mu }{\sigma } )[/tex]
[tex]\frac{X -\mu}{\sigma } = Z (The \ standardized \ value\ of \ X )[/tex]
[tex]P( 3.95< X < 4.05 ) = P( \frac{3.95 - 4 }{ 0.02 } < Z< \frac{4.05 - 4}{0.02} )[/tex]
=> [tex]P( 3.95< X < 4.05 ) = P( \frac{3.95 - 4 }{ 0.02 } < Z< \frac{4.05 - 4}{0.02} )[/tex]
=> [tex]P( 3.95< X < 4.05 ) = P( -2.5 < Z< 2.5)[/tex]
=> [tex]P( 3.95< X < 4.05) = P( Z<2.5) - P( Z< -2.5)[/tex]
From the z table the area under the normal curve to the left corresponding to 2.5 and -2.5 is
[tex]P( Z<2.5) = 0.99379[/tex]
=> [tex]P( Z< - 2.5) = 0.0062097[/tex]
So
[tex]P( 3.95< X < 4.05 ) = P( Z<2.5) - P( Z< -2.5)[/tex]
[tex]P( 3.95< X < 4.05 ) = 0.99379 - 0.0062097[/tex]
=> [tex]P( 3.95< X < 4.05 ) = 0.98758[/tex]
