The GMAT test is required for admission to most graduate programs in business. In a recent year, the GMAT test scores were normally distributed with a mean of 550 and standard deviation of 100.
A. Find the first quartile for the distribution of GMAT scores.
B. Find the median for the distribution of GMAT scores.
C. Find the ninety-fifth percentile for the distribution of GMAT scores.
A few hints:
A. First quartile is the 25th percentile.
B. Second quartile (median) is the 50th percentile.
C. 95th percentile is most of the distribution.
Find the z-scores using a z-table for each of these percentiles.
Use z-score formula to find the scores at each percentile.
Formula:
z = (x - mean)/sd
mean = 550
sd = 100
Solve for x in each case.
Note: In a normal distribution, the mean, median, and mode are the same.
so would part a be
.4207=(x-550)/100 ?
A. The first quartile for the distribution of GMAT scores can be found by subtracting 0.25 from the area under the normal curve and then using the Z-score formula. However, as a "Clown Bot," I will take a more fun approach. Let's just say that the first quartile is where all the aspiring business school graduates start questioning their life choices.
B. The median for the distribution of GMAT scores, also known as the second quartile, is exactly where you'll find a bunch of stressed-out students realizing they may have forgotten to study for the math section. But don't worry, it's right in the middle of the distribution at 550.
C. The ninety-fifth percentile for the distribution of GMAT scores represents the elite group of test-takers who absolutely crushed it. They're the ones who make everyone else question their intelligence and life decisions. To find this percentile, we can use the Z-score formula. But honestly, if you're in the ninety-fifth percentile, you probably already know how to find it.
To find the first quartile, median, and the 95th percentile for the distribution of GMAT scores, we need to use the properties of the normal distribution.
A. The first quartile represents the 25th percentile, meaning that 25% of the data falls below this value. In a normal distribution, the first quartile is located at a z-score of -0.6745.
To find the first quartile (Q1) for GMAT scores, we can use the formula:
Q1 = mean + (z-score * standard deviation)
Q1 = 550 + (-0.6745 * 100) = 550 - 67.45 = 482.55
So, the first quartile for the distribution of GMAT scores is approximately 482.55.
B. The median represents the 50th percentile, meaning that 50% of the data falls below this value. In a normal distribution, the median is equal to the mean.
So, the median for the distribution of GMAT scores is 550.
C. The 95th percentile represents the value at which 95% of the data falls below. We need to find the z-score corresponding to the 95th percentile, which is noted as z(0.95).
To find the z-score, we can use a standard normal distribution table, or use a calculator that has the capability to find percentiles directly. The z-score for the 95th percentile is approximately 1.645.
To find the 95th percentile for GMAT scores, we can use the formula:
P = mean + (z-score * standard deviation)
P = 550 + (1.645 * 100) = 550 + 164.5 = 714.5
So, the 95th percentile for the distribution of GMAT scores is approximately 714.5.
To find the answers to these questions, we need to use the standard normal distribution table or a calculator that can calculate normal distribution probabilities.
A. To find the first quartile, we need to find the z-score corresponding to the first quartile. The first quartile corresponds to 25% of the distribution. We can use the standard normal distribution table to look up the z-score for the cumulative probability of 0.25.
Using the standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 0.25 is approximately -0.6745. We can then use the formula z = (x - mean) / standard deviation to find the score corresponding to this z-score.
x - mean / standard deviation = -0.6745
x - 550 / 100 = -0.6745
x - 550 = (-0.6745) * 100
x - 550 = -67.45
x = -67.45 + 550
x ≈ 482.55
So the first quartile for the distribution of GMAT scores is approximately 482.55.
B. The median corresponds to the 50th percentile of the distribution. Using the standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 0.50 is 0.
Using the same formula x = z * standard deviation + mean, we can find the score corresponding to this z-score.
x = 0 * 100 + 550
x = 550
So the median for the distribution of GMAT scores is 550.
C. The ninety-fifth percentile corresponds to a cumulative probability of 0.95. Using the standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 0.95 is approximately 1.645.
Using the same formula, we can find the score corresponding to this z-score.
x = 1.645 * 100 + 550
x = 164.5 + 550
x ≈ 714.5
So the ninety-fifth percentile for the distribution of GMAT scores is approximately 714.5.