a normal distribution ha am ean of 80 with a standard deviation of 20. what score separates the highest 15% of the distribution from the rest of the score?
the formula of z-score:
z=(x-mu)/sigma
where mu=mean and sigma=standard deviation
find the z-score equivalent on the table that is equivalent to .85 to substitute back into equation and solve for x
Well, well, well, looks like someone needs help dividing the cool kids from the rest of the bunch in a normal distribution. So, what you're after is the score that separates the highest 15% from the rest, huh?
Now, in a normal distribution, we have this fancy thing called the z-score. The z-score tells us how many standard deviations a data point is away from the mean. And lucky for us, we have calculators and tables to figure these things out.
So, to find the magical score that separates the top 15% from the rest, we need to find the z-score value that corresponds to that percentile. And let me tell you, the z-score's wardrobe is full of surprises!
Using a z-table or calculator, we can find that the z-score that corresponds to the highest 15% (or a percentile of 0.85) is approximately 1.04.
Now, we can use this z-score to figure out the actual score by utilizing the z-score formula: z = (X - mean) / standard deviation.
So, plugging in the numbers, we have 1.04 = (X - 80) / 20.
Now we release our inner algebra wizard and solve for X: X = 1.04 * 20 + 80.
Drumroll, please! After some mathematical juggling, we discover that the score separating the highest 15% of the distribution from the rest is approximately 100.8.
So, there you have it! If you manage to score higher than 100.8, you'll be part of the "elite" club. Good luck, my friend!
To find the score that separates the highest 15% of the distribution from the rest of the scores, you can use the Z-score.
1. First, find the Z-score corresponding to the cumulative probability of 0.85 (which represents the highest 15% of the distribution). This can be done using a Z-table or a calculator.
2. From the Z-table, the closest Z-score to 0.85 is approximately 1.04.
3. The formula for calculating the Z-score is:
Z = (X - μ) / σ
where Z is the Z-score, X is the score you want to find, μ is the mean, and σ is the standard deviation.
4. Rearranging the formula to solve for X, we get:
X = Z * σ + μ
5. Substitute the known values into the formula:
X = 1.04 * 20 + 80
6. Calculate the result:
X = 20.8 + 80
X ≈ 100.8
Therefore, a score of approximately 100.8 separates the highest 15% of the distribution from the rest of the scores.
To find the score that separates the highest 15% of the distribution from the rest, you need to determine the z-score at the 85th percentile of a standard normal distribution. Here's how you can calculate it:
Step 1: Standardize the given mean and standard deviation by using the z-score formula: z = (x - mean) / standard deviation.
In this case, the mean (μ) is 80 and the standard deviation (σ) is 20.
So, z = (x - 80) / 20.
Step 2: Look up the z-score that corresponds to the 85th percentile. Since we are interested in the highest 15%, we subtract 85 from 100 to get the value for the upper tail: 100 - 85 = 15.
Step 3: Find the z-score associated with the upper tail area of 0.15 using a standard normal distribution table or a statistical calculator. The result will be positive since we are looking at the upper tail.
Step 4: Substitute the z-score value into the z-score formula and solve for x.
(x - 80) / 20 = z-score
Solve for x:
x = (z-score * standard deviation) + mean
Substitute the value of the z-score in the equation and calculate:
x = (z-score * 20) + 80
This will give you the score that separates the highest 15% of the distribution from the rest.
Note: Statistical tables or calculators can provide the z-score corresponding to the 85th percentile. It is approximately 1.036. So, substituting this value into the equation:
x = (1.036 * 20) + 80
x ≈ 101.73
Therefore, a score of approximately 101.73 separates the highest 15% of the distribution from the rest of the scores.