assume that adults have IQ scores that are normally distributed with a mean of 105 and a standard deviation of 15 find the probability that a randomly selected adult has an IQ between 89 and 121 ?
Convert to z-score using
(X-μ)/σ
where X=89 or 121, σ=15, μ=105.
Look up normal distribution table to find probability enclosed between these two z-scores.
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To find the probability that a randomly selected adult has an IQ between 89 and 121, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Standardize the values
We need to convert the raw IQ scores into z-scores (standard scores) using the formula:
z = (x - μ) / σ
where x is the value, μ is the mean, and σ is the standard deviation.
For 89:
z1 = (89 - 105) / 15
= -16 / 15
= -1.07
For 121:
z2 = (121 - 105) / 15
= 16 / 15
= 1.07
Step 2: Find the area under the curve
Next, we use a standard normal distribution table, calculator, or software to find the area under the curve between z1 and z2. Since the normal distribution is symmetric, the area between z1 and z2 is the same as the area between -z1 and -z2.
Looking up the z-values in a standard normal distribution table, we find that the area to the left of -1.07 is 0.1423, and the area to the left of 1.07 is 0.8577.
Thus, the area between z1 and z2 is:
0.8577 - 0.1423 = 0.7154
Therefore, the probability that a randomly selected adult has an IQ between 89 and 121 is approximately 0.7154, or 71.54%.
To find the probability that a randomly selected adult has an IQ between 89 and 121, we can use the properties of the normal distribution.
Step 1: Standardize the scores
First, we need to standardize the values of 89 and 121 using the formula for standardizing a score:
z = (x - μ) / σ
where z is the standardized score, x is the given value, μ is the mean, and σ is the standard deviation.
For 89:
z = (89 - 105) / 15 = -16 / 15 = -1.067
For 121:
z = (121 - 105) / 15 = 16 / 15 = 1.067
Step 2: Find the probabilities associated with these standardized scores
Next, we can use a z-table or a calculator to find the probability associated with each standardized score.
Using a z-table or calculator, the probability associated with -1.067 is approximately 0.1423, and the probability associated with 1.067 is also approximately 0.1423.
Step 3: Calculate the probability between the two scores
To find the probability between the two scores, we can subtract the probability associated with the lower score from the probability associated with the higher score:
P(89 ≤ X ≤ 121) = P(X ≤ 121) - P(X ≤ 89)
P(X ≤ 121) = 0.1423
P(X ≤ 89) = 0.1423
P(89 ≤ X ≤ 121) = 0.1423 - 0.1423 = 0
Therefore, the probability that a randomly selected adult has an IQ between 89 and 121 is approximately 0.