Some IQ scores are standardized with a mean of 100 and a standard deviation of 16. Using the 68-95-99.7 Rule, Determine: In what interval you would expect the middle 95% of the IQ scores to be found.
For a normal distribution, 68.27% of members lie within 1 s.d. of the mean, 95.45% within 2 s.d., and 99.73% within 3 s.d.
They want you to round these numbers off to 68-95-99.7, so: 95% of scores are within 2 standard deviations of the mean (100). Therefore the answer is 68 to 132
To determine the interval in which you would expect the middle 95% of the IQ scores to be found, we can use the 68-95-99.7 Rule, also known as the Empirical Rule or Three Sigma Rule.
According to this rule, for a normal distribution:
- Approximately 68% of the observations fall within one standard deviation of the mean,
- Approximately 95% of the observations fall within two standard deviations of the mean,
- Approximately 99.7% of the observations fall within three standard deviations of the mean.
In your case, the IQ scores are standardized with a mean of 100 and a standard deviation of 16. Since we are interested in the middle 95%, which corresponds to two standard deviations, we can calculate the upper and lower bounds using the following formula:
Lower bound = Mean - (2 * Standard Deviation)
Upper bound = Mean + (2 * Standard Deviation)
Plugging in the values, we get:
Lower bound = 100 - (2 * 16) = 100 - 32 = 68
Upper bound = 100 + (2 * 16) = 100 + 32 = 132
Therefore, you would expect the middle 95% of the IQ scores to be found within the interval [68, 132].