X is normally dsitributed with mean 250 and standard deviation 40. what value of X does only top 15% exceed?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.15) and its Z score. Insert all values in above equation to calculate the score.
To find the value of X that exceeds only the top 15%, we need to determine the z-score corresponding to this percentile and then convert it back to the original X value.
The z-score represents the number of standard deviations a data point is from the mean. The z-score can be calculated using the formula:
z = (X - μ) / σ
where X is the value we want to find, μ is the mean, and σ is the standard deviation.
To find the z-score for the top 15% (or the area under the curve), we need to subtract this value from 1 since the area under the curve represents the cumulative probability up to that point.
So, the z-score for the top 15% is:
z = invNorm(1 - 0.15) [using a z-table or inverse normal distribution function]
Now, we can substitute the values into the formula to solve for X:
(X - 250) / 40 = z
X - 250 = 40z
X = 40z + 250
Substituting the value of z we found earlier, we can get the value of X:
X = 40( invNorm(1 - 0.15) ) + 250
Using a calculator or statistical software, we can find the value of invNorm(1 - 0.15) to be approximately 1.036.
X = 40(1.036) + 250
X ≈ 290.44
Therefore, the value of X that only the top 15% exceeds is approximately 290.44.
To find the value of X that only the top 15% exceed, we need to use the standard normal distribution.
1. Convert the given values to the standard normal distribution by using the formula: z = (x - μ) / σ, where z is the standard score, x is the observed value, μ is the mean, and σ is the standard deviation.
For the given problem:
μ = 250 (mean)
σ = 40 (standard deviation)
2. Use a standard normal distribution table or a statistical calculator to find the z-score that corresponds to the top 15% of the distribution. Since we want to find the value that only the top 15% exceed, we need to find the z-score corresponding to the 85th percentile.
3. Look up the z-score of 0.85 in the standard normal distribution table or use a statistical calculator. The z-score corresponding to the 85th percentile is approximately 1.0364.
4. Solve for x in the formula z = (x - μ) / σ:
1.0364 = (x - 250) / 40
5. Rearrange the equation to solve for x:
40 * 1.0364 = x - 250
41.456 = x - 250
x = 41.456 + 250
x ≈ 291.456
Therefore, the value of X that only the top 15% exceed is approximately 291.456.