During a study of the IQ of Idahoans researchers find that a

During a study of the IQ of Idahoans, researchers find that all scores have amound shaped distribution with mean IQ of 100 and variance 81. Draw the relative frequency distribution and determine the range of most IQ scores in Idaho.

a) Calculate the percentage of Idahoans who have IQ\'s between 82 and 118. Define and calculate the 50th percentile.

b) If my ex-wife has an IQ of 73 how many Idahoans suffer the same difficulites in their day to day affairs as she?

c) My friend Pete from Pocatello has an IQ of 150. Calculate Pete\'s z-score and determine if he is a genius. When should I use the Theorem and when would I be able to use the Empirical rule to calculate these percentages.

Solution

During a study of the IQ of Idahoans, researchers find that all scores have amound shaped distribution with mean IQ of 100 and variance 81

Mound shaped distribution is also known as Normal distribution.

mean = 100

variance = 81

sd = sqrt(variance) = sqrt(81) = 9

a) Calculate the percentage of Idahoans who have IQ\'s between 82 and 118.

By using Emperical rule,

It states that the 68% of the data falls within first standard deviation,

the 95% falls within first two standard deviations and

99.7% of the data falls within first three standard deviations of the mean.

µ + = 100 + 9 = 109

µ - = 100 - 9 = 91

µ + 2* = 100 + (2*9) = 118

µ - 2* = 100 - (2*9) = 82

µ + 3* = 100 + (3*9) = 127

µ - 3* = 100 - (3*9) = 73

the percentage of Idahoans who have IQ\'s between 82 and 118 is 95%.

Define and calculate the 50th percentile.

The formula below is used to compute percentiles of a normal distribution.

X = µ + z*

That means the area = 50/100 = 0.5

We have to find z-score when area is 0.5.

By using statistical table,

z = 0.00

X = 100 + (0.00*9) = 100

The 50th percentile is 100.

b) If my ex-wife has an IQ of 73 how many Idahoans suffer the same difficulites in their day to day affairs as she?

We are already calculated that

µ - 3* = 100 - (3*9) = 73

that means IQ of 73 is below three standard deviation from the mean.

c) My friend Pete from Pocatello has an IQ of 150. Calculate Pete\'s z-score and determine if he is a genius. When should I use the Theorem and when would I be able to use the Empirical rule to calculate these percentages.

We are given that X = 150 , mean = 100 and sd = 9

Z = (X - mean) / sd

= (150 - 100) / 9 = 5.556