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jeka94
3 years ago
8

A population of monkeys' tail lengths is normally distributed with a mean of 25 cm with a standard deviation of 8 cm. I am prepa

ring to take a sample of size 256 from this population, and record the tail length of each monkey in my sample. What is the probability that the mean of my sample will be between 24 and 25 cm?
Mathematics
1 answer:
Crank3 years ago
4 0

Answer:

<em>The probability that the mean of my sample will be between 24 and 25 cm</em>

<em>P(24 ≤X⁻≤25) = 0.4772</em>

Step-by-step explanation:

<u><em>Step(i)</em></u>:-

<em>Given mean of the Population  'μ'= 25c.m</em>

<em>Given standard deviation of the Population 'σ' = 8c.m</em>

<em>Given sample size 'n' = 256</em>

<em>Let X₁ = 24</em>

<em></em>Z_{1}  = \frac{x_{1}-mean }{\frac{S.D}{\sqrt{n} } } = \frac{24-25}{\frac{8}{\sqrt{256} } } =  -2<em></em>

<em>Let X₂ = 25</em>

<em></em>Z_{2}  = \frac{x_{2}-mean }{\frac{S.D}{\sqrt{n} } } = \frac{25-25}{\frac{8}{\sqrt{256} } } = 0<em></em>

<u><em>Step(ii)</em></u><em>:-</em>

<em>The probability that the mean of my sample will be between 24 and 25 cm</em>

<em>P(24 ≤X⁻≤25) = P(-2≤ Z ≤0)</em>

                     = P( Z≤0) - P(Z≤-2)

                     = 0.5 + A(0) - (0.5- A(-2))

                     = A(0) + A(2)        ( ∵A(-2) =A(2)

                     = 0.000+ 0.4772

                     = 0.4772

<u><em>Final answer</em></u>:-

<em>The probability that the mean of my sample will be between 24 and 25 cm</em>

<em>P(24 ≤X⁻≤25) = 0.4772</em>

                     

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