Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 12
.
Use the empirical rule to determine the following.
(a) What percentage of people has an IQ score between 76 and 124?
(b) What percentage of people has an IQ score less than 88 or greater than 112?
(c) What percentage of people has an IQ score greater than 124?
To solve these questions using the empirical rule, we need to know that the empirical rule states that in a bell-shaped distribution:
- 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.
(a) The range between 76 and 124 is two standard deviations away from the mean (100).
So, approximately 95% of the people will have an IQ score between 76 and 124.
(b) To find the percentage of people with an IQ score less than 88 or greater than 112, we need to determine the proportion of data that is outside two standard deviations (less than 88 or greater than 112).
Since the empirical rule states that approximately 95% falls within two standard deviations, approximately (100% - 95%)/2 = 2.5% of the data will fall outside two standard deviations.
So, approximately 2.5% of the people will have an IQ score less than 88 or greater than 112.
(c) To find the percentage of people with an IQ score greater than 124, we need to determine the proportion of data that is greater than two standard deviations above the mean (greater than 112).
Since the empirical rule states that approximately 95% falls within two standard deviations, approximately (100% - 95%)/2 = 2.5% of the data will fall outside two standard deviations (greater than 112).
So, approximately 2.5% of the people will have an IQ score greater than 124.
To solve these questions using the empirical rule, also known as the 68-95-99.7 rule, we will assume that the IQ scores follow a bell-shaped distribution.
The rule states that:
- 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.
Given that the mean is 100 and the standard deviation is 12, we can use these values to find the requested percentages.
(a) What percentage of people has an IQ score between 76 and 124?
To find the percentage between these two scores, we need to determine the range within two standard deviations of the mean.
Lower limit = Mean - (2 * Standard Deviation) = 100 - (2 * 12) = 76
Upper limit = Mean + (2 * Standard Deviation) = 100 + (2 * 12) = 124
Therefore, approximately 95% of people have an IQ score between 76 and 124.
(b) What percentage of people has an IQ score less than 88 or greater than 112?
To find the percentage for this question, we need to determine the range beyond one standard deviation from the mean (+/-), excluding the range between 88 and 112.
Lower limit = Mean - (1 * Standard Deviation) = 100 - 12 = 88
Upper limit = Mean + (1 * Standard Deviation) = 100 + 12 = 112
The range between 88 and 112 is within one standard deviation from the mean. So, the range outside this interval is the complement of the percentage within the range, which is 100% - 68% = 32%.
Therefore, approximately 32% of people have an IQ score less than 88 or greater than 112.
(c) What percentage of people has an IQ score greater than 124?
To find the percentage for this question, we need to determine the range beyond two standard deviations from the mean.
Upper limit = Mean + (2 * Standard Deviation) = 100 + (2 * 12) = 124
The range beyond two standard deviations from the mean is the complement of the percentage within this range, which is 100% - 95% = 5%.
Therefore, approximately 5% of people have an IQ score greater than 124.