Find the area of the shaded region.The graph to the right depicts IQ scores of Adults (85-120), and those scores are normally distributed with a mean of 100 and a standard deviation of 15
you can play around a lot with Z table stuff here:
http://davidmlane.com/hyperstat/z_table.html
Another method:
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability between the two Z scores.
To find the area of the shaded region, we need to calculate the probability that an individual has an IQ score within a certain range. In this case, we want to find the probability of having an IQ score between 85 and 120.
Step 1: Standardize the given IQ scores
To standardize an IQ score, we use the formula z = (x - μ) / σ, where z is the standard score, x is the raw score, μ is the mean, and σ is the standard deviation.
For an IQ score of 85:
z1 = (85 - 100) / 15 = -1
For an IQ score of 120:
z2 = (120 - 100) / 15 = 1.33
Step 2: Find the corresponding probabilities
The next step is to find the probability of having an IQ score less than or equal to 85 (P(X ≤ 85)) and the probability of having an IQ score less than or equal to 120 (P(X ≤ 120)).
Using a standard normal distribution table or a calculator, we can find the corresponding probabilities:
P(X ≤ 85) = 0.1587 (approx.)
P(X ≤ 120) = 0.9088 (approx.)
Step 3: Calculate the area of the shaded region
To find the area of the shaded region, we subtract the probability of having an IQ score less than or equal to 85 from the probability of having an IQ score less than or equal to 120:
Area = P(85 ≤ X ≤ 120) = P(X ≤ 120) - P(X ≤ 85) = 0.9088 - 0.1587 = 0.7501 (approx.)
Therefore, the area of the shaded region is approximately 0.7501, or 75.01%.
To find the area of the shaded region, we need to determine the probability associated with that area. Since IQ scores are normally distributed with a mean of 100 and a standard deviation of 15, we can use the standard normal distribution to find the probability.
1. First, let's draw a normal distribution curve with a mean of 100 and a standard deviation of 15.
2. Next, we need to identify the specific range of IQ scores that corresponds to the shaded region. From the graph, it appears that the shaded region represents scores below 85 or above 120.
3. To calculate the area of the shaded region, we need to find the cumulative probability of both tails - the area less than 85 and the area greater than 120.
4. We can use the Z-score formula to convert our IQ scores into standard scores (Z-scores) which are values on the standard normal distribution.
Z = (X - μ) / σ
where X is the value (IQ score), μ is the mean, and σ is the standard deviation.
For X = 85:
Z1 = (85 - 100) / 15 = -1
For X = 120:
Z2 = (120 - 100) / 15 = 1.33
5. Now, we can use a standard normal distribution table, or a calculator, to find the cumulative probability associated with these Z-scores.
P(Z < -1) = 0.1587 (approximately)
P(Z > 1.33) = 0.0918 (approximately)
6. Since the shaded region includes both tails, we need to add the probabilities of the two tails to find the total probability.
Total probability = 0.1587 + 0.0918 = 0.2505 (approximately)
7. Finally, to find the area of the shaded region, we subtract the total probability from 1.
Area of shaded region = 1 - 0.2505 = 0.7495 (approximately)
Therefore, the area of the shaded region is approximately 0.7495, which corresponds to a probability of 74.95%.