This distribution is very close to Normal with a mean of 111 and a standard devi

Question

This distribution is very close to Normal with a mean of 111 and a standard deviation of 11. It includes all seventh graders in a rural Midwest school except for 4 low outliers who were dropped because they may have been ill or otherwise not paying attention to the test. Consider the Normal distribution with a mean of 111 and a standard deviation of 11 as a description of the IQ test scores of all rural Midwest seventh-grade students.

Fill in the blank. Use the 68-95-99.7 Rule to complete the following sentence: ____ percent of all students have IQ scores below 78? (Give your answer to two decimal places.)

Explanation / Answer

Given that the mean (u) of a normal probability distribution is 111 and the
standard deviation (sd) is 11                                  
About 99.7% of the area under the normal curve is within two standard deviation of the mean. i.e. (u ± 3s.d)
So to the given normal distribution about 99.7% of the observations lie in between
= [111 ± 3 * 11]
= [ 111 - 3 * 11 , 111 + 3* 11]
= [ 78 , 144 ]                  

percent of all students have IQ scores below 78 = ( 1 - (0.997) )/2 = 0.0015
0.15% of students have IQ scores below 78

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