Question

The scores on an IQ test follow the normal curve quite closely. The average score was 100 . 40% of the people scored between 100 points and 113 points. What percentage of people scored between 100 points and 120 points?

Answer 1

Approximately 40.8% of the people scored between 100 points and 120 points on the IQ test.

Step-by-step explanation:

AI-generated answer

The percentage of people who scored between 100 points and 120 points on the IQ test can be calculated using the properties of the normal curve.

Since the average score on the IQ test is 100, we know that 50% of the people scored below 100 points and 50% scored above 100 points.

From the given information, we know that 40% of the people scored between 100 points and 113 points. This means that 40% of the people scored between the average score (100 points) and 113 points.

To find the percentage of people who scored between 100 points and 120 points, we need to calculate the area under the normal curve between these two scores.

To do this, we need to find the z-scores corresponding to these two scores. The z-score represents the number of standard deviations a score is from the mean.

First, we calculate the z-score for 100 points:

z-score = (100 - mean) / standard deviation

Since the normal curve follows the 68-95-99.7 rule, we know that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Let's assume that the standard deviation is 15 (this is just an example, as the question does not provide the actual standard deviation).

Using the formula, the z-score for 100 points would be:

z-score = (100 - 100) / 15 = 0

Next, we calculate the z-score for 120 points:

z-score = (120 - mean) / standard deviation

z-score = (120 - 100) / 15 = 1.33 (rounded to two decimal places)

To find the percentage of people who scored between 100 points and 120 points, we need to find the area under the normal curve between these two z-scores.

Using a standard normal distribution table or a calculator, we can find that the area to the left of a z-score of 0 is 0.5 and the area to the left of a z-score of 1.33 is approximately 0.908 (rounded to three decimal places).

To find the area between these two z-scores, we subtract the smaller area from the larger area:

0.908 - 0.5 = 0.408 (rounded to three decimal places)

Therefore, approximately 40.8% of the people scored between 100 points and 120 points on the IQ test.

Answer 2

Approximately 40.82% of people scored between 100 points and 120 points on the IQ test.

Step-by-step explanation:

The percentage of people who scored between 100 points and 120 points can be found by calculating the area under the normal curve between these two points.

First, we need to find the z-scores for these two points using the formula:

z = (x - μ) / σ

where x is the score, μ is the mean, and σ is the standard deviation.

For the first point, z = (100 - 100) / 15 = 0.

For the second point, z = (120 - 100) / 15 = 1.33.

Next, we can use a standard normal table or a calculator to find the area under the curve between these two z-scores.

From the standard normal table, the area between z = 0 and z = 1.33 is approximately 0.4082 or 40.82%.

Therefore, approximately 40.82% of people scored between 100 points and 120 points on the IQ test.