Question

a) choose the model for these iq scores that correctly shows what the 68 - 95 - 99.7 rule predicts about the scores.
b) in what interval would you expect the central 68% of the iq scores to be found?
using the 68 - 95 - 99.7 rule, the central 68% of the iq scores are between 84 and 116.
(type integers or decimals. do not round.)
c) about what percent of people should have iq scores above 148?
using the 68 - 95 - 99.7 rule, about.15 % of people should have iq scores above 148.
(type an integer or a decimal. do not round.)
d) about what percent of people should have iq scores between 68 and 84?
using the 68 - 95 - 99.7 rule, about % of people should have iq scores between 68 and 84.
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule for a normal distribution states that about 68% of the data lies within 1 standard - deviation ($\mu\pm\sigma$) of the mean, about 95% lies within 2 standard - deviations ($\mu\pm2\sigma$) of the mean, and about 99.7% lies within 3 standard - deviations ($\mu\pm3\sigma$) of the mean.

Step2: Analyze part a

The correct model for the 68 - 95 - 99.7 rule has 68% of the data in the $\mu - \sigma$ to $\mu+\sigma$ interval, 95% in the $\mu - 2\sigma$ to $\mu + 2\sigma$ interval, and 99.7% in the $\mu - 3\sigma$ to $\mu+3\sigma$ interval. The correct model is C.

Step3: Analyze part b

For a normal distribution of IQ scores with mean $\mu = 100$ and standard deviation $\sigma = 16$, the central 68% of the data lies in the interval $\mu-\sigma$ to $\mu+\sigma$. Substituting the values, we get $100 - 16=84$ and $100 + 16 = 116$.

Step4: Analyze part c

The value 148 is $\mu+3\sigma$ (since $\mu = 100$ and $\sigma = 16$, $100+3\times16=148$). The total area under the normal curve is 100%. The area within $\mu - 3\sigma$ to $\mu+3\sigma$ is 99.7%. The area above $\mu + 3\sigma$ is $\frac{100 - 99.7}{2}=0.15$.

Step5: Analyze part d

The value 68 is $\mu - 2\sigma$ ($100-2\times16 = 68$) and 84 is $\mu-\sigma$ ($100 - 16=84$). The area within $\mu - 2\sigma$ to $\mu - \sigma$ is half of the difference between the area within $\mu - 2\sigma$ to $\mu+2\sigma$ (95%) and the area within $\mu - \sigma$ to $\mu+\sigma$ (68%). So, $\frac{95 - 68}{2}=13.5$.

Answer:

a) C.
b) 84, 116
c) 0.15
d) 13.5