Suppose that IQ scores have a bell-shaped distribution with a mean of 103 and a standard deviation of 14. Using the empirical rule, what percentage of IQ scores are at least 145? Please do not round your answer.
To find the percentage of IQ scores that are at least 145 using the empirical rule, we can determine the number of standard deviations above the mean that 145 falls.
First, we calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value (145 in this case), μ is the mean (103), and σ is the standard deviation (14).
z = (145 - 103) / 14
z = 42 / 14
z = 3
The value 145 is 3 standard deviations above the mean.
Next, we refer to the empirical rule which states:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
Since 145 falls beyond three standard deviations above the mean, we can conclude that approximately 99.7% of the IQ scores are lower than or equal to 145.
Therefore, the percentage of IQ scores that are at least 145 is approximately 100% - 99.7% = 0.3%.