Question

A population of values has a normal distribution with
μ=65.3μ=65.3 and σ=42.4σ=42.4. You intend to draw a random sample
of size n=242n=242.

Find *P*_{34}, which is the mean separating the
bottom 34% means from the top 66% means.

*P*_{34} (for sample means) =

Enter your answers as numbers accurate to 1 decimal place. Answers
obtained using exact *z*-scores or *z*-scores rounded
to 3 decimal places are accepted.

Answer #1

Given, the population is normally distributed, X ~ N( )
= N(65.3, 42.4^{2})

Now, when we draw a sample of size n = 242 from this population, the sample is also normally distributed with,

Mean, = 65.3, and std deviation, s = / sqrt(n) = 42.4 / sqt(242) = 2.726

Hence, the sample has a normal distribution, Y ~ N(65.3,
2.726^{2})

Hence, P_{34} = 34th percentile of the sample
distribution = y (say)

=> P(Y <= y) = 0.34

=> P(Z <= (y-65.3) / 2.726 ) = 0.34 (converting to std Z-score probability)

=> (y - 65.3) / 2.726 = -0.41

=> y = 64.18

Hence, **P _{34} = 64.18**

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Enter your answers as numbers accurate to 1 decimal place. Answers
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1Info Details
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