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?
Accepted Solution
A:
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 = 100standard deviation = 13According to the empirical rule68% data lies between [tex]\overline{x}-\sigma[/tex] and [tex]\overline{x}+\sigma[/tex].95% data lies between [tex]\overline{x}-2\sigma[/tex] and [tex]\overline{x}+2\sigma[/tex].99.7% data lies between [tex]\overline{x}-3\sigma[/tex] and [tex]\overline{x}+3\sigma[/tex].(a)[tex]\overline{x}-\sigma=100-13=87[/tex][tex]\overline{x}+\sigma=100+13=113[/tex][tex][\overline{x}-\sigma,\overline{x}+\sigma]=[87,113][/tex]Using empirical rule we can say that 68% of people has an IQ score between 87 and 113.(b)[tex]\overline{x}-2\sigma=100-2(13)=74[/tex][tex]\overline{x}+2\sigma=100+2(13)=126[/tex][tex][\overline{x}-2\sigma,\overline{x}+2\sigma]=[74,126][/tex]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 isP = 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)[tex]\overline{x}-3\sigma=100-3(13)=61[/tex][tex]\overline{x}+3\sigma=100+3(13)=139[/tex][tex][\overline{x}-3\sigma,\overline{x}+3\sigma]=[61,139][/tex]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 isP = 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[tex]P=\frac{1}{2}(0.3\%)=0.15\%[/tex]Therefore 0.15% of people has an IQ score greater than 139.