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lina2011 [118]
3 years ago
14

A population of values has a normal distribution with μ = 157.7 and σ = 57 . You intend to draw a random sample of size n = 24 .

Find P95, which is the mean separating the bottom 95% means from the top 5% means.
Mathematics
1 answer:
alexgriva [62]3 years ago
5 0

Answer:

z=1.64

And if we solve for a we got

a=157.7 +1.64*57=251.18

So the value of height that separates the bottom 95% of data from the top 5% is 251.18.  

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the variable of interest of a population, and for this case we know the distribution for X is given by:

X \sim N(157.7,57)  

Where \mu=157.7 and \sigma=57

For this part we want to find a value a, such that we satisfy this condition:

P(X>a)=0.05   (a)

P(X   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.95 of the area on the left and 0.05 of the area on the right it's z=1.64. On this case P(Z<1.64)=0.95 and P(z>1.64)=0.05

If we use condition (b) from previous we have this:

P(X  

P(z

But we know which value of z satisfy the previous equation so then we can do this:

z=1.64

And if we solve for a we got

a=157.7 +1.64*57=251.18

So the value of height that separates the bottom 95% of data from the top 5% is 251.18.  

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