Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 13. Use the empirical rule to determine the following. ​(a) What percentage of people has an IQ score between 87 and 113​? ​(b) What percentage of people has an IQ score less than 74 or greater than 126​? ​(c) What percentage of people has an IQ score greater than 139​?

Answer 1

Answer:

(a) 68% of people has an IQ score between 87 and 113​.

(b) 5% of people has an IQ score less than 74 or greater than 126.

(c) 0.15% of people has an IQ score greater than 139​.

Step-by-step explanation:

Given information:Scores of an IQ test have a​ bell-shaped distribution,

mean = 100

standard deviation = 13

According to the empirical rule

68% data lies between $\overline{x}-\sigma$ and $\overline{x}+\sigma$.

95% data lies between $\overline{x}-2\sigma$ and $\overline{x}+2\sigma$.

99.7% data lies between $\overline{x}-3\sigma$ and $\overline{x}+3\sigma$.

(a)

$\overline{x}-\sigma=100-13=87$

$\overline{x}+\sigma=100+13=113$

$[\overline{x}-\sigma,\overline{x}+\sigma]=[87,113]$

Using empirical rule we can say that 68% of people has an IQ score between 87 and 113​.

(b)

$\overline{x}-2\sigma=100-2(13)=74$

$\overline{x}+2\sigma=100+2(13)=126$

$[\overline{x}-2\sigma,\overline{x}+2\sigma]=[74,126]$

Using empirical rule we can say that 95% of people has an IQ score between 74 and 126​.

The percentage of people has an IQ score less than 74 or greater than 126 is

P = 1- percent of people has an IQ score between 74 and 126​.

P = 1- 95%

P = 5%

Therefore 5% of people has an IQ score less than 74 or greater than 126.

(c)

$\overline{x}-3\sigma=100-3(13)=61$

$\overline{x}+3\sigma=100+3(13)=139$

$[\overline{x}-3\sigma,\overline{x}+3\sigma]=[61,139]$

Using empirical rule we can say that 99.7% of people has an IQ score between 61 and 139​.

The percentage of people has an IQ score less than 61 or greater than 139 is

P = 1- percent of people has an IQ score between 61 and 139​.

P = 1- 99.7%

P = 0.3%

The percentage of people has an IQ score greater than 139​ is

$P=\frac{1}{2}(0.3\%)=0.15\%$

Therefore 0.15% of people has an IQ score greater than 139​.

Answer 2

68% percent of people has an IQ score between 87 and 113, 5% percent of people has an IQ score less than 74 or greater than 126,​ and 0.15% percent of people has an IQ score greater than 139 and this can be determined by using the empirical rule.

Given :

Scores of an IQ test have a​ bell-shaped distribution with a mean of 100 and a standard deviation of 13.

Empirical rule said that:

68% of data lies between $\rm \bar{x} - \sigma$ and $\rm \bar{x} + \sigma$.

95% of data lies between $\rm \bar{x} - 2\sigma$ and $\rm \bar{x} + 2\sigma$.

99.7% of data lies between $\rm \bar{x} - 3\sigma$ and $\rm \bar{x} + 3\sigma$.

A)

$\rm \bar{x} - \sigma$ = 100 - 13 = 87

$\rm \bar{x} + \sigma$ = 100 + 13 = 113

[$\rm \bar{x} - \sigma$ , $\rm \bar{x} + \sigma$] = [87 , 113]

68% of people have an IQ score between 87 and 113.

B)

$\rm \bar{x} - 2\sigma$ = 100 - 26 = 74

$\rm \bar{x} + 2\sigma$ = 100 + 26 = 126

[$\rm \bar{x} - 2\sigma$ , $\rm \bar{x} + 2\sigma$] = [74 , 126]

95% of people have an IQ score between 74 and 126.

The percentage of people who have an IQ score less than 74 or greater than 126 is:

P = 100 - 95

P = 5%

C)

$\rm \bar{x} - 3\sigma$ = 100 - 39 = 61

$\rm \bar{x} + 3\sigma$ = 100 + 39 = 139

[$\rm \bar{x} - 3\sigma$ , $\rm \bar{x} + 3\sigma$] = [61 , 139]

99.7% of people have an IQ score between 61 and 139.

The percentage of people who have an IQ score less than 61 or greater than 139 is:

P = 100 - 99.7

P = 0.3%

The percentage of people who have an IQ score greater than 139​ is:

P = 0.5 (0.3)

P = 0.15%

0.15% percentage of people has an IQ score greater than 139.

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https://brainly.com/question/23017717