Suppose that IQ scores have a bell-shaped distribution with a mean of 97 and a standard deviation of 17. Using the empirical rule, what percentage of IQ scores are no more than 46?
Correct answer is approximately 0.15% of IQ scores are no more than 46.
To find the percentage of IQ scores that are no more than 46 using the empirical rule, we need to calculate the z-score for the given value and then use the z-score table to determine the percentage.
The z-score formula is:
z = (x - mean) / standard deviation
In this case, the mean (µ) is 97, the standard deviation (σ) is 17, and the value (x) is 46.
Substituting these values into the formula, we get:
z = (46 - 97) / 17
z ≈ -3
Now we need to find the percentage associated with this z-score. Consulting a z-score table, we find that an approximate z-score of -3 corresponds to a cumulative area of 0.0013.
This means that approximately 0.13% of IQ scores are no more than 46.
So, the percentage of IQ scores that are no more than 46 using the empirical rule is 0.13%.
To answer this question using the empirical rule, we need to apply the concept of standard deviations and the percentages commonly associated with them.
The empirical rule states that for data that follows a normal distribution (or a bell-shaped curve), approximately:
- 68% of the data falls within one standard deviation of the mean,
- 95% of the data falls within two standard deviations of the mean,
- 99.7% of the data falls within three standard deviations of the mean.
First, we need to determine how far away 46 is from the mean. We calculate the number of standard deviations by subtracting the mean from 46 and dividing by the standard deviation:
z = (46 - 97) / 17
z = -51 / 17
z ≈ -3
Since we have a negative value, it means that 46 is 3 standard deviations below the mean on the left side of the bell curve.
According to the empirical rule, we know that 99.7% of the data falls within three standard deviations of the mean. Therefore, the percentage of IQ scores that are no more than 46 is approximately 99.7%.
46 is three standard deviations below the mean
approx. 0.13% of the scores are at or below this point