Scores of an IQ test have a bell-shaped distribution with a mean of 100 and a standard deviation of 17. What percentage of people has an in score between 83 and 117
To find the percentage of people with an IQ score between 83 and 117, we need to find the area under the bell-shaped curve between these two scores.
First, we need to standardize the scores by subtracting the mean from each score and dividing by the standard deviation.
For a score of 83:
Standardized score = (83 - 100) / 17 = -1
For a score of 117:
Standardized score = (117 - 100) / 17 = 1
Now, we can find the area under the curve between these standardized scores using a standard normal distribution table or calculator.
Using a standard normal distribution table, we can find the area for a standardized score of -1 as 0.1587. This represents the percentage of people below a score of 83.
Similarly, the area for a standardized score of 1 is 0.8413, representing the percentage of people below a score of 117.
To find the area between these two scores, we subtract the area below 83 from the area below 117:
0.8413 - 0.1587 = 0.6826
So approximately 68.26% of people have an IQ score between 83 and 117.
To find the percentage of people with scores between 83 and 117, we need to find the area under the normal distribution curve between these two values.
First, we need to standardize the values using the z-score formula:
z = (x - mean) / standard deviation
For 83:
z = (83 - 100) / 17 = -17 / 17 = -1
For 117:
z = (117 - 100) / 17 = 17 / 17 = 1
Next, we use a z-table or a calculator to find the area under the curve between -1 and 1.
The z-table gives us the area to the left of a given z-score. To find the area between -1 and 1, we can subtract the area to the left of -1 from the area to the left of 1.
Using the z-table, the area to the left of -1 is 0.1587, and the area to the left of 1 is 0.8413.
So, the area between -1 and 1 is:
0.8413 - 0.1587 = 0.6826
This means that approximately 68.26% of people have an IQ score between 83 and 117.
To find the percentage of people with a score between 83 and 117, you need to calculate the area under the bell-shaped curve (also known as a normal distribution) between those two scores.
First, convert the scores to z-scores using the formula:
z = (X - μ) / σ
where X is the score, μ is the mean, and σ is the standard deviation.
For 83:
z1 = (83 - 100) / 17 = -1.0
For 117:
z2 = (117 - 100) / 17 = 1.0
Now, we can use a z-score table or a calculator to find the area under the curve between these two z-scores.
Using a standard normal distribution table, you can find that the area to the left of z = -1.0 is approximately 0.1587, and the area to the left of z = 1.0 is approximately 0.8413.
To find the area between these two z-scores, subtract the area to the left of z1 from the area to the left of z2:
Area = 0.8413 - 0.1587 = 0.6826
Finally, convert this area to a percentage by multiplying it by 100:
Percentage = 0.6826 * 100 = 68.26%
Therefore, approximately 68.26% of the people have an IQ score between 83 and 117.