A normal distribution has a mean of 120 and a standard deviation of 20. What range of scores would form the middle 60%?
To find the range of scores that would form the middle 60% of a normal distribution, we will use the property that the middle range of a normal distribution falls within one standard deviation of the mean in each direction.
Step 1: Calculate the z-scores corresponding to the middle 60% of the distribution.
The middle 60% equates to the range between the 20th and 80th percentiles. We can find the corresponding z-scores using a standard normal distribution table or calculator.
Since the distribution is symmetric, we can calculate the z-scores for the lower and upper boundaries as follows:
Lower z-score: (100% - 60%) / 2 = 20%
Upper z-score: (100% + 60%) / 2 = 80%
Using a standard normal distribution table or calculator, we find that the z-score corresponding to the 20th percentile is approximately -0.8416, and the z-score corresponding to the 80th percentile is approximately 0.8416.
Step 2: Convert the z-scores back to raw scores.
To convert the z-scores to raw scores, we use the formula: X = mean + (z * standard deviation)
Lower score = 120 + (-0.8416 * 20) = 120 - 16.83 ≈ 103.17
Upper score = 120 + (0.8416 * 20) = 120 + 16.83 ≈ 136.83
Therefore, the range of scores that would form the middle 60% of the normal distribution is approximately 103.17 to 136.83.
To find the range of scores that would form the middle 60% of a normal distribution, you need to calculate the z-scores for the lower and upper limits.
Step 1: Find the z-score for the lower limit:
To find the z-score corresponding to the lower limit, you need to determine the percentile associated with it. The middle 60% means that 30% is on either side of the middle. So, you would subtract 30% from 50% to find the lower limit's percentile: 50% - 30% = 20%.
The corresponding z-score for the 20th percentile can be found using a z-table or a statistical calculator. For the z-table, you look up the closest value to 20% (0.20) in the table, which is approximately 0.84. So, the z-score for the lower limit is -0.84 (because it falls on the left side of the mean in a standard normal distribution).
Step 2: Find the z-score for the upper limit:
The upper limit's percentile can be found by adding 30% to the 50% midpoint. Therefore, the upper limit's percentile is 80%.
Using the z-table once again, you find the closest value to 80% (0.80), which is approximately 0.84. So, the z-score for the upper limit is also 0.84 (because it falls on the right side of the mean in a standard normal distribution).
Step 3: Convert z-scores to raw scores:
Now that you have the z-scores for the lower and upper limits, you can calculate the corresponding raw scores using the formula: X = μ + (z * σ), where X is the raw score, μ is the mean, z is the z-score, and σ is the standard deviation.
For the lower limit:
X = 120 + (-0.84 * 20) = 120 - 16.8 = 103.2
For the upper limit:
X = 120 + (0.84 * 20) = 120 + 16.8 = 136.8
Thus, the range of scores that would form the middle 60% is from approximately 103.2 to 136.8.