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A population of values has a normal distribution with u = 232 and o = 21.4. A random sample of size n = 85is drawn. Find the probability that a sample of size n = 85 is randomly selected with a mean greater than 235.9.Round your answer to four decimal places.PM > 235.9)

Answer :

Given:

[tex]\mu=232,\text{ }\sigma=21.4,\text{ n=85 and M=235.9}[/tex]

The z-score value is

[tex]z=\frac{M-\mu}{\frac{\sigma}{\sqrt[]{n}}}[/tex]

[tex]\text{ Substitue }\mu=232,\text{ }\sigma=21.4,\text{ n=85 and M=235.9, we get}[/tex]

[tex]z=\frac{235.9-232}{\frac{21.4}{\sqrt[]{85}}}[/tex]

[tex]z=\frac{\sqrt[]{85}(235.9-232)}{21.4}[/tex]

[tex]z=1.68019[/tex]

P value from the z table is

[tex]P(-235.9M<235.9)=0.45354[/tex][tex]P(M<235.9)-0.5=0.45354[/tex]

[tex]P(M<235.9)=0.95354[/tex]

We know that

[tex]P(M>235.9)=1-P(M<235.9)[/tex]

[tex]P(M>235.9)=1-0.95354[/tex]

[tex]P(M>235.9)=0.046459[/tex]

Hence the answer is

[tex]P(M>235.9)=0.046459[/tex]

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