Question

Question Help Scores of an IQ test have a​ bell-shaped distribution with a mean of 100100 and a standard deviation of 1010. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 8080 and 120120​? ​(b) What percentage of people has an IQ score less than 9090 or greater than 110110​? ​(c) What percentage of people has an IQ score greater than 120120​?

Answer 1

Answer:

a) 95% percentage of people has an IQ score between 80 and 120.

b) 68% percentage of people has an IQ score between 90 and 110.    

c) 2.5% percentage of people has an IQ score greater than 120.          

Step-by-step explanation:

The empirical formula states that:

• The empirical rule is known as the three-sigma rule or 68-95-99.7 rule.
• It is a rule which states that for a normal distribution, almost all data falls within three standard deviations of the mean.
• 68% of data falls within the first standard deviation that is $\mu \pm \sigma$

• 95% of data falls within the second standard deviation that is $\mu \pm 2\sigma$

• 99.7% of data falls within the third standard deviation that is $\mu \pm 3\sigma$

a)  percentage of people has an IQ score between 80 and 120

(80, 120) can be written as:

$(80, 120) = (100-2(10), 100 + 2(10))\\= (\mu \pm 2\sigma)$

Thus, it is the interval within two standard deviation from the mean.

Thus, by empirical formula, 95% percentage of people has an IQ score between 80 and 120.

b) percentage of people has an IQ score between 90 and 110

(90, 110) can be written as:

$(90, 110) = (100-1(10), 100 + 1(10))\\= (\mu \pm 1\sigma)$

Thus, it is the interval within one standard deviation from the mean.

Thus, by empirical formula, 68% percentage of people has an IQ score between 90 and 110.

c) P(score greater than 120)

= 1 - percentage of score less than 80 - percentage of score between 80 and 120

$= 1 - (95\% + 2.5\%) = 2.5\%$

From empirical formula,

2.5% percentage of people has an IQ score greater than 120.