Question

In psychology, there is a particular Mental Development Index (MDI) used in the study of infants. The scores on the MDI have approximately a normal distribution with a mean of 100 and standard deviation of 16. We are going to randomly select 64 children and average their MDI scores. What is the probability that the average is under 105?

Give your answer to 4 decimal places. For help on how to input a numeric answer, please see "Instructions for inputting a numeric response."

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Answer 1

The question can be broken down into an approximation for the probability of finding <105.

For a normal distribution, it is possible to convert our into a corresponding z value which can then be used to assign the probability of that z value. A z table can be used to acquire this z value. A z table is a tabulation of the values obtained from the probability distribution normal. We assume a normal curve with mean 0 and standard deviation 1. When this curve is standardized we can use the z table to navigate the difficult data provided we have a sufficiently large sample.

Here,

=100

=16

n=64

we have to find P(<105).

The general conversion equation is

By substituting the appropriate values the equation becomes

and by applying the values provided to us we get. It is a good idea to organize the given data as above.

Our z value has been found to be 2.5.

P(<105) becomes P(z<2.5)

to get the probability of getting z value lesser than 2.5 we take the value corresponding to 2.5 on the left z-table.

z=2.5 corresponds to P = 0.9938.

The probability of getting a mean MDI lesser than 105 is 0.9938.

The z table can be used to find the probability of a range of values in the same way.