on the assumption that IQ's are normally distribited in the population with a mean of 100 and a standard deviation of 20 findbthe proportion of peoples IQ's. what is the probability that a score picked at random will lie above score of 135, lie below score 90, what is the probability of scoring between 90 and 120, lie above score 120
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 for Z in each of the situations.
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To find the proportions and probabilities, we can use the standard normal distribution since we are given the mean and standard deviation. The standard normal distribution has a mean of 0 and a standard deviation of 1.
To convert the given values to the standard normal distribution, we use the formula:
Z = (X - μ) / σ
Where:
Z is the standard score (z-score)
X is the value
μ is the mean
σ is the standard deviation
1. Probability of IQ score above 135:
To calculate this, we need to find the area under the standard normal curve to the right of 135. We calculate the z-score for 135:
Z = (135 - 100) / 20 = 1.75
Using a standard normal table or a calculator, we find the proportion of values to the right of Z = 1.75. In this case, the probability is around 0.0401, or approximately 4.01%.
2. Probability of IQ score below 90:
To calculate this, we need to find the area under the standard normal curve to the left of 90. We calculate the z-score for 90:
Z = (90 - 100) / 20 = -0.5
Using a standard normal table or a calculator, we find the proportion of values to the left of Z = -0.5. In this case, the probability is around 0.3085, or approximately 30.85%.
3. Probability of scoring between 90 and 120:
We first find the z-scores for both 90 and 120:
Z₁ = (90 - 100) / 20 = -0.5
Z₂ = (120 - 100) / 20 = 1
Next, we calculate the area to the left of Z₂ minus the area to the left of Z₁:
P = P(Z₁ < Z < Z₂) = P(Z < 1) - P(Z < -0.5)
Using a standard normal table or a calculator, we find that P(Z < 1) is approximately 0.8413 and P(Z < -0.5) is approximately 0.3085. Subtracting these values gives us the probability of approximately 0.5328, or approximately 53.28%.
4. Probability of IQ score above 120:
To calculate this, we need to find the area under the standard normal curve to the right of 120. We calculate the z-score for 120:
Z = (120 - 100) / 20 = 1
Using a standard normal table or a calculator, we find the proportion of values to the right of Z = 1. In this case, the probability is around 0.1587, or approximately 15.87%.
So, the probabilities are as follows:
- Probability of IQ score above 135: 0.0401 or 4.01%
- Probability of IQ score below 90: 0.3085 or 30.85%
- Probability of scoring between 90 and 120: 0.5328 or 53.28%
- Probability of IQ score above 120: 0.1587 or 15.87%