Scores of an IQ test have a bell shaped distribution with a mean of 100 and a standard deviation of 19. Use the empirical rule to determine the following.
a.) What % of people has an IQ score between 81 and 119?
To determine the percentage of people with IQ scores between 81 and 119, we can use the empirical rule. According to the empirical rule, approximately 68% of the data falls within one standard deviation of the mean, approximately 95% falls within two standard deviations, and approximately 99.7% falls within three standard deviations.
First, let's calculate the z-scores for the IQ scores of 81 and 119. The z-score can be calculated using the formula (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
For an IQ score of 81:
z = (81 - 100) / 19
z ≈ -0.95
For an IQ score of 119:
z = (119 - 100) / 19
z ≈ 1
Now, let's find the percentage of people with IQ scores between these two z-scores. Since the data is normally distributed, we can use the area under the curve to determine the percentage.
Using a standard normal distribution table or a calculator, the approximate percentage of people with IQ scores between -0.95 and 1 is:
Area between z = -0.95 and z = 1 ≈ 68%
Therefore, approximately 68% of people have an IQ score between 81 and 119.
To determine the percentage of people with an IQ score between 81 and 119, we can use the empirical rule, also known as the 68-95-99.7 rule. This rule states that for a bell-shaped distribution (also known as a normal distribution), 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
In this case, the mean IQ score is 100 and the standard deviation is 19. Let's calculate the values for one standard deviation, two standard deviations, and three standard deviations from the mean:
One standard deviation below the mean: 100 - 19 = 81
One standard deviation above the mean: 100 + 19 = 119
Thus, an IQ score between 81 and 119 falls within one standard deviation of the mean. According to the empirical rule, approximately 68% of people will have an IQ score within this range.
Z score for 81 is (81-100)/19=-1
Z score for 119 is (119-100)/19=+1
Use normal distribution table to find the difference in probability between -1 and +1. It should be around 68%.