A particular normal distribution has mean of 575 and standard deviation of 150.
What X values form the boundaries for the middle 60% of the data? Round answers to two decimal places.
Lower boundary: Blank 1
Upper Boundary: Blank 2
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.30 from mean) and get the Z scores.
Insert Z scores into equation below.
Z = (score-mean)/SD
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To find the boundaries for the middle 60% of the data, we need to determine the values that cut off 20% on each tail of the normal distribution.
Step 1: Find the Z-scores corresponding to the cutoff points
Since the normal distribution is standardized with a mean of 0 and a standard deviation of 1, we can use Z-scores to find the cutoff points.
To find the Z-score that corresponds to the lower cutoff point, use the formula:
Z = (X - μ) / σ
Substituting the given values, we have:
Z = (X - 575) / 150
To find the Z-score that corresponds to the upper cutoff point, we have the same formula:
Z = (X - μ) / σ
Step 2: Find the Z-scores corresponding to the cutoff percentages
Since we want to cut off 20% on each tail, the remaining 60% lies in the middle. This means the cutoff points will be at the 20th and 80th percentiles.
To find the Z-scores corresponding to these percentiles, we can use a Z-table or a calculator.
Using a Z-table, we can find that the Z-score corresponding to the 20th percentile is approximately -0.84, and the Z-score corresponding to the 80th percentile is approximately 0.84.
Step 3: Solve for the X values
By equating the Z-scores to the Z formula and solving for X, we can find the X values that correspond to the cutoff points.
For the lower boundary:
-0.84 = (X - 575) / 150
Solving this equation, we find:
X - 575 = -0.84 * 150
X - 575 = -126
X = 575 - 126
X ≈ 449.00
For the upper boundary:
0.84 = (X - 575) / 150
Solving this equation, we find:
X - 575 = 0.84 * 150
X - 575 = 126
X = 575 + 126
X ≈ 701.00
Therefore, the boundaries for the middle 60% of the data are:
Lower boundary: 449.00
Upper boundary: 701.00