Find the area of the shaded region. the graph to the right depicts iq scores of adults and those scores are normally distributed with a mean of 100 and a standard deviation of 15. the shaded area 83
To find the area of the shaded region, we need to use the properties of the normal distribution and its associated Z-scores.
Step 1: Convert the shaded area to a Z-score.
The Z-score represents the number of standard deviations an observation is from the mean. We can find the Z-score corresponding to the shaded area using a Z-table or a calculator with a built-in normal distribution function. Since we want to find the right-tail area, we subtract the shaded area from 1, giving us 1 - 0.83 = 0.17.
Step 2: Find the Z-score value.
The Z-table provides the area to the left of a given Z-score. In this case, we need to find the Z-score associated with an area of 0.17 to the left. By looking up the Z-score in the table or using the calculator's inverse normal distribution function, we can find the Z-score.
Step 3: Convert the Z-score back to the actual IQ score.
Using the Z-score from step 2, we can convert it back to the original IQ score using the formula: IQ = (Z-score * standard deviation) + mean. In this case, the mean is 100 and the standard deviation is 15.
Once we have the IQ score corresponding to the Z-score, we can use it to interpret the result or further analysis, such as identifying individuals within that range or comparing it to different groups.
Note that the equation for converting Z-scores to IQ scores assumes a normal distribution of IQ scores.
To find the area of the shaded region, we need to compute the z-score corresponding to the specified area.
The z-score, also known as the standard score, represents the number of standard deviations an observation or data point is from the mean. In this case, we want to find the z-score corresponding to the area of 83%.
To find the z-score, we need to use the cumulative distribution function (CDF) of the standard normal distribution. However, the given information does not provide the exact value of the z-score for an area of 83%. Therefore, we will use the concept of symmetry to find the corresponding z-score.
Since the area under the normal distribution curve is symmetric, we can find the z-score for an area of 0.5 + (1 - 0.83) = 0.335, which is a total of 33.5% on both sides of the mean.
Using a standard normal distribution table or a calculator, we find that the z-score for an area of 0.335 is approximately 0.43.
Now that we have the z-score, we can use it to find the corresponding raw score using the formula:
X = Mean + (Z-score * Standard Deviation)
Substituting the values into the formula:
X = 100 + (0.43 * 15)
X = 100 + 6.45
X ≈ 106.45
Therefore, the raw score corresponding to a z-score of approximately 0.43 is 106.45.
Finally, since we are looking for the area to the right of this raw score (which includes the shaded region), we can subtract this area from 1 to find the shaded area:
Shaded area = 1 - 0.835
Shaded area ≈ 0.165 (or 16.5%)
Hence, the area of the shaded region is approximately 16.5%.