Question regarding a normal distribution chart: ten thousand adults were given a literacy test. the results were nearly normally distributed (μ = 75, σ = 15).

a. about how many scored between 60 and 80?

b. approximately what score did only the top 15% exceed?

c. find the percentile rank of the person who scored 88.

a)

So then we expect aroudn 0.470*10000= 4700 adults scored between 60 and 80.

b)

So the score that separates the bottom 85% of data from the top 15% is 90.6.  

c)

So the score that separates the bottom 88% of data from the top 12% is 92.55

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".  

Part a

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:

Where and

We are interested on this probability

And the best way to solve this problem is using the normal standard distribution and the z score given by:

If we apply this formula to our probability we got this:

And we can find this probability on this way:

So then we expect aroudn 0.470*10000= 4700 adults scored between 60 and 80.

Part b

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

  (a)

  (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 0.85 of the area on the left and 0.15 of the area on the right it's z=1.04. On this case P(Z<1.04)=0.85 and P(z>1.04)=0.15

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

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

And if we solve for a we got

So the score that separates the bottom 85% of data from the top 15% is 90.6.

Part c

For this case we want to find a value that satisfy these two conditions:

  (a)

  (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 0.88 of the area on the left and 0.12 of the area on the right it's z=1.17. On this case P(Z<1.17)=0.88 and P(z>1.17)=0.12

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

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

And if we solve for a we got

So the score that separates the bottom 88% of data from the top 12% is 92.55

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