Assume that adults have iq scores that are normally distributed with a mean of 100 and a standard deviation of 15 as on the Wechsler test find the iq score separating the top 14% from the others
Use table in back of stat book to find Z for top 14%, then use equation below.
Z = (score - mean)/ SD
To find the IQ score that separates the top 14% from the others, we need to find the z-score associated with the 14th percentile and then convert it back to an IQ score using the mean and standard deviation.
1. First, we need to find the z-score associated with the 14th percentile. We can use a standard normal distribution table or a calculator to find this value.
The z-score corresponding to the 14th percentile is approximately -1.08.
2. Next, we will use the formula for z-score conversion to find the IQ score corresponding to the z-score:
z = (X - μ) / σ
Rearranging the formula to solve for X, we get:
X = z * σ + μ
Plugging in the values, we get:
X = -1.08 * 15 + 100
Solving this equation:
X = -16.2 + 100
X ≈ 83.8
Therefore, the IQ score separating the top 14% from the others is approximately 83.8.
To find the IQ score separating the top 14% from the others, we need to use the standard normal distribution table or a statistical calculator.
Here's how you can find the IQ score:
Step 1: Convert the given probability (14%) to a z-score.
- As we know that the normal distribution is symmetrical, we need to find the z-score that corresponds to the upper tail of 100% - 14% = 86%.
- The z-score corresponding to the upper tail of 0.86 can be obtained from the standard normal distribution table or by using a calculator.
Step 2: Calculate the z-score by finding the inverse of the cumulative distribution function (CDF) using a standard normal distribution table or a calculator.
- Using a standard normal distribution table, the closest value to 0.86 is 0.8159.
- This means that the z-score corresponding to 86% is approximately 0.8159.
Step 3: Convert the z-score back to the IQ score using the formula:
IQ score = (z-score * standard deviation) + mean
Given:
Mean (μ) = 100
Standard Deviation (σ) = 15
IQ score = (0.8159 * 15) + 100
IQ score ≈ 112.24
Therefore, the IQ score separating the top 14% from the others on the Wechsler test is approximately 112.24.