Question

a particular iq test is standardized to a normal model, with a mean of 100 and a stan

b) in what interval would you expect the central 99.7% of the iq scores to be found?
using the 68 - 95 - 99.7 rule, the central 99.7% of the iq scores are between \\( \square \\) and \\( \square \\).
(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 \\( \square \\% \\) 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 \\( \square \\% \\) of people should have iq scores between 68 and 84.
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall the 68 - 95 - 99.7 rule for normal distribution

The 68 - 95 - 99.7 rule states that for a normal distribution:

• Approximately 68% of the data lies within \( \mu\pm\sigma\)
• Approximately 95% of the data lies within \( \mu\pm2\sigma\)
• Approximately 99.7% of the data lies within \( \mu\pm3\sigma\)

Given \( \mu = 100\).
Assume \( \sigma=16\) (since \(100 + 3\times16=148\) and \(100- 3\times16 = 52\), \(100+2\times16 = 132\), \(100 - 2\times16=68\), \(100+1\times16 = 116\), \(100-1\times16 = 84\))

Step2: Solve part (b)

For the central 99.7% of the data (using \( \mu\pm3\sigma\)):
\( \mu - 3\sigma=100-3\times16=100 - 48 = 52\)
\( \mu+3\sigma=100 + 3\times16=100+48 = 148\)

Step3: Solve part (c)

The total area under the normal curve is 100%. The area within \( \mu\pm3\sigma\) is 99.7%. The area outside \( \mu\pm3\sigma\) is \(100 - 99.7=0.3\%\). Since the normal distribution is symmetric, the area above \( \mu + 3\sigma\) is \( \frac{100 - 99.7}{2}=0.15\%\)

Step4: Solve part (d)

The area within \( \mu\pm1\sigma\) is 68% and the area within \( \mu\pm2\sigma\) is 95%.
The area within \( \mu - 2\sigma\) and \( \mu-\sigma\) is \(\frac{95 - 68}{2}=13.5\%\)

Answer:

b) \(52\) and \(148\)
c) \(0.15\)
d) \(13.5\)