A Population follows a normal distribution with mean 10 and standard deviation 2. In a sample of 100, what is the probability of the sample mean being between 9.5 and 10.1?

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A Population follows a normal distribution with mean 10 and standard deviation 2. In a sample of 100, what is the probability of the sample mean being between 9.5 and 10.1?

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Answer 1 · 2017-01-19T06:02:04

Probability of the sample mean being between 9.5 and 10.1 $= 0.5859$ $=0.5859$ or 90%

Explanation:

Given -

Population Mean $= 100$ $= 100$
Population Standard Deviation $= 2$ $=2$

Standard Error $= \frac{σ}{\sqrt{n}} = \frac{2}{\sqrt{100}} = \frac{2}{10} = 0.2$ $=sigma/sqrtn=2/sqrt100=2/10=0.2$

Find $z$ $z$ for $μ = 9.5, μ = 10.5$ $mu=9.5, mu=10.5$

$z = \frac{μ - μ_{¯ x}}{S E}$ $z=(mu -mu_(barx))/(SE)$

At $x = 9.5; z = \frac{9.5 - 10}{0.2} = \frac{- 0.5}{0.2} = - 1.25$ $x=9.5; z = (9.5-10)/0.2=(-0.5)/0.2=-1.25$
At $x = 9.5; z = \frac{10.1 - 10}{0.2} = \frac{0.1}{0.2} = 0.5$ $x=9.5; z = (10.1-10)/0.2=(0.1)/0.2=0.5$
Area between $z = - 0.25 and z = 0.05 = 0.3944 + 0.01915 = 0.5859$ $z=-0.25 and z= 0.05=0.3944 +0.01915=0.5859$

Probability of the sample mean being between 9.5 and 10.1 $= 0.5859$ $=0.5859$ or 90%

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A Population follows a normal distribution with mean 10 and standard deviation 2. In a sample of 100, what is the probability of the sample mean being between 9.5 and 10.1?

1 Answer

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