Some IQ scores are standardized with a mean of 100 and a standard deviation of 16. Using the
68-95-99.7 Rule, Determine:
a.In what interval you would expect the middle 95% of the IQ scores to be found.
b.What percent of people would have an IQ score above 116.
c.What percent of people would have an IQ score 68 and 84.
d.What percent of people would have an IQ score of at least 132.
e.What percent of people would have an IQ score of no more than 68.
f.What percent of people would have an IQ score of 100 or above.
Determine what?
To answer these questions, we can use the concept of the 68-95-99.7 Rule, also known as the empirical rule or the three-sigma rule. This rule states that in 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
a. In what interval would you expect the middle 95% of the IQ scores to be found?
To find the interval where the middle 95% of the IQ scores would be found, we need to identify the range within two standard deviations of the mean. Since the mean is 100 and the standard deviation is 16, we can calculate the range as follows:
Lower limit: mean - (2 * standard deviation) = 100 - (2 * 16) = 100 - 32 = 68
Upper limit: mean + (2 * standard deviation) = 100 + (2 * 16) = 100 + 32 = 132
Therefore, the middle 95% of the IQ scores would be found within the interval 68 to 132.
b. What percent of people would have an IQ score above 116?
To find the percentage of people with an IQ score above 116, we need to determine how many standard deviations above the mean 116 is:
Z-score = (IQ score - mean) / standard deviation
Z-score = (116 - 100) / 16 = 1
We can then find the percentage of people by looking at the area to the right of the mean in the standard normal distribution table corresponding to a Z-score of 1. The table tells us that this area is approximately 0.1587, or 15.87%.
Therefore, approximately 15.87% of people would have an IQ score above 116.
c. What percent of people would have an IQ score between 68 and 84?
To find the percentage of people with an IQ score between 68 and 84, we need to determine how many standard deviations each score is from the mean:
Z-score for 68 = (68 - 100) / 16 = -2
Z-score for 84 = (84 - 100) / 16 = -1
We can then find the percentage of people by finding the area between these two Z-scores in the standard normal distribution table. By referring to the table, we find that the area between -1 and -2 is approximately 0.1359, or 13.59%.
Therefore, approximately 13.59% of people would have an IQ score between 68 and 84.
d. What percent of people would have an IQ score of at least 132?
To find the percentage of people with an IQ score of at least 132, we need to determine how many standard deviations above the mean 132 is:
Z-score = (IQ score - mean) / standard deviation
Z-score = (132 - 100) / 16 = 2
From the standard normal distribution table, we can find the area to the right of the mean corresponding to a Z-score of 2, which is approximately 0.0228, or 2.28%.
Therefore, approximately 2.28% of people would have an IQ score of at least 132.
e. What percent of people would have an IQ score of no more than 68?
To find the percentage of people with an IQ score of no more than 68, we need to determine how many standard deviations below the mean 68 is:
Z-score = (IQ score - mean) / standard deviation
Z-score = (68 - 100) / 16 = -2
From the standard normal distribution table, we can find the area to the left of the mean corresponding to a Z-score of -2, which is approximately 0.0228, or 2.28%.
Therefore, approximately 2.28% of people would have an IQ score of no more than 68.
f. What percent of people would have an IQ score of 100 or above?
To find the percentage of people with an IQ score of 100 or above, we need to determine how many standard deviations above the mean 100 is:
Z-score = (IQ score - mean) / standard deviation
Z-score = (100 - 100) / 16 = 0
From the standard normal distribution table, we can find the area to the right of the mean corresponding to a Z-score of 0, which is approximately 0.5, or 50%.
Therefore, approximately 50% of people would have an IQ score of 100 or above.