X is normally distributed with a mean of 250 and a standard deviation of 40. What value of X does only the 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 and get its Z score. Insert values into equation above and solve for the score.
112.8
To find the value of X that is exceeded by only the top 15%, we need to determine the z-score corresponding to the area of 0.15 in the upper tail of the normal distribution.
The z-score can be calculated using the formula: z = (X - μ) / σ, where X represents the value, μ is the mean, and σ is the standard deviation.
First, let's find the z-score corresponding to the top 15% in the upper tail. We can use a table of standard normal probabilities or a calculator to find this value.
The area to the left of the z-score of interest is equal to 1 - 0.15 = 0.85. Looking up the z-score for an area of 0.85 in the standard normal distribution table, we find that the corresponding z-score is approximately 1.036.
Now that we have the z-score, we can rearrange the formula: z = (X - μ) / σ, to solve for X:
1.036 = (X - 250) / 40
Next, we can solve for X by multiplying both sides of the equation by 40 and then adding 250:
1.036 * 40 = X - 250
41.44 = X - 250
X = 41.44 + 250
X ≈ 291.44
Therefore, the value of X that is exceeded by only the top 15% is approximately 291.44.