Find the area of the shaded region. The graph depicts the IQ score of adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.
The shaded region numbers are 85-120
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To find the area of the shaded region, which represents the probability of an adult having an IQ score between 85 and 120, we can use the properties of a normal distribution.
Here's how you can calculate it:
Step 1: Standardize the values
To work with a normal distribution, we need to standardize the values using the formula:
z = (x - μ) / σ
Where:
- x is the value you want to standardize (in this case, 85 and 120)
- μ is the mean of the distribution (given as 100)
- σ is the standard deviation of the distribution (given as 15)
For the value 85, the standardized value will be:
z1 = (85 - 100) / 15
Similarly, for the value 120, the standardized value will be:
z2 = (120 - 100) / 15
Step 2: Find the cumulative probabilities
Once we have the standardized values, we can find the cumulative probabilities associated with those values. This gives us the area under the normal curve up to those points.
Using a standard normal distribution table, you can look up the cumulative probabilities associated with the standardized values z1 and z2. These tables provide the probability values for a standard normal distribution, which has a mean of 0 and a standard deviation of 1.
For example, if we find the cumulative probability value for z1 (as shown in the standard normal distribution table), it represents the area under the curve up to the z-score of 85. Let's call this value P1.
Similarly, finding the cumulative probability value for z2 (from the table) represents the area under the curve up to the z-score of 120. Let's call this value P2.
Step 3: Calculate the shaded area
To find the area of the shaded region (the probability of an IQ score between 85 and 120), we subtract P1 from P2.
Area of the shaded region = P2 - P1
This gives us the probability of an adult having an IQ score between 85 and 120.
Remember to refer to a standard normal distribution table or a statistical software tool to find the exact cumulative probabilities and calculate the area of the shaded region.