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 % of people should have iq scores above 148. (type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the 68 - 95 - 99.7 rule

The 68 - 95 - 99.7 rule for a normal distribution states that about 68% of the data is within 1 standard - deviation ($\mu\pm\sigma$) of the mean, about 95% is within 2 standard - deviations ($\mu\pm2\sigma$) of the mean, and about 99.7% is within 3 standard - deviations ($\mu\pm3\sigma$) of the mean. The mean of IQ scores is $\mu = 100$ and the standard deviation is $\sigma = 16$.

Step2: Analyze part (a)

The correct model should have 68% within $\mu\pm\sigma$ (i.e., 84 - 116), 95% within $\mu\pm2\sigma$ (i.e., 68 - 132) and 99.7% within $\mu\pm3\sigma$ (i.e., 52 - 148). Option A is correct.

Step3: Analyze part (b)

As mentioned above, for a normal distribution of IQ scores with $\mu = 100$ and $\sigma = 16$, the central 68% of the data is in the interval $\mu\pm\sigma=100 - 16$ to $100 + 16$, which is 84 to 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%. Since 99.7% of the data is within $\mu\pm3\sigma$, the remaining area outside of $\mu\pm3\sigma$ is $100 - 99.7=0.3\%$. This area is split evenly between the two tails. So the area above $\mu + 3\sigma$ (i.e., above 148) is $\frac{100 - 99.7}{2}=0.15\%$.

Answer:

a) A.
b) 84, 116
c) 0.15