A government report looked at the amount borrowed for college by students who graduated in 2000 and had taken out student loans. The mean amount was = $ 17 , 776 and the standard deviation was s = $12,034. The quartiles were Q 1 = $9900, M = $15,532, and Q 3 = $22,500.
The right skew pulls the standard deviation up. So a Normal distribution with the same mean and standard deviation would have a third quartile larger than the actual Q 3. Find the third quartile of the Normal distribution with μ = $17,776 and σ = $12,034 and compare it with Q 3 = $22,500.
i know the answer is Q3 would be $25,838.78. This is a lot larger than our value of Q3 ($22,500).. but how what do i do to get that answer? thanks
To find the third quartile (Q3) of the normal distribution, we would use the Z-score formula:
Z = (X - μ) / σ
where X is the value we're interested in (Q3 in this case), μ is the mean, and σ is the standard deviation. In this problem, we have μ = $17,776 and σ = $12,034.
For Q3 in a normal distribution, we know that 75% of the data falls below that value. We can use a Z-table or calculator to find the Z-score that corresponds to 75% of the data. This Z-score is approximately 0.67.
Now, we'll use the Z-score formula above to find the value of Q3:
0.67 = (Q3 - 17776) / 12034
Now, solve for Q3:
0.67 * 12034 = Q3 - 17776
8022.78 = Q3 - 17776
Q3 = 8022.78 + 17776
Q3 = $25,798.78
So, the third quartile of the normal distribution with μ = $17,776 and σ = $12,034 is approximately $25,798.78. This is larger than the given Q3 value of $22,500, which confirms what we were expecting.
To find the third quartile (Q3) of a Normal distribution with a given mean (μ) and standard deviation (σ), you can use z-scores.
First, calculate the z-score corresponding to the value of Q3. The z-score formula is:
z = (x - μ) / σ,
where x is the value you want to convert to a z-score. In this case, x = Q3 = $22,500, μ = $17,776, and σ = $12,034.
z = ($22,500 - $17,776) / $12,034
z = $4,724 / $12,034
z ≈ 0.3927
Next, look up the z-score in the standard normal distribution table or use a calculator that can calculate normal probabilities. The z-score of 0.3927 corresponds to a cumulative probability of approximately 0.6515.
Since Q3 corresponds to the third quartile, which is the 75th percentile, subtract the cumulative probability from 1:
1 - 0.6515 = 0.3485.
Now find the value (x) that corresponds to this cumulative probability of 0.3485 from the standard normal distribution table or use a calculator. In this case, x ≈ 0.3907.
Lastly, convert the z-score back to a value by rearranging the z-score formula:
z = (x - μ) / σ,
x - μ = z * σ,
x = z * σ + μ.
Plugging in the values, you get:
x = 0.3907 * $12,034 + $17,776
x ≈ $25,838.78.
So, the third quartile (Q3) of the Normal distribution with μ = $17,776 and σ = $12,034 is approximately $25,838.78. This value is larger than the actual Q3 value of $22,500, indicating a right skew in the distribution.
To find the third quartile (Q3) of a normal distribution with a given mean (μ) and standard deviation (σ), you will use the concept of z-scores.
The formula to calculate the z-score for a specific value (x) in a normal distribution is:
z = (x - μ) / σ
In this case, you want to find the z-score for Q3. The z-score formula can be rearranged to find x:
x = z * σ + μ
Since you want to find Q3, which is 75% of the data, the corresponding z-score is the one that separates the top 25% of the distribution from the bottom 75%. This z-score can be found using a standard normal distribution table or a statistical calculator.
By looking up a z-table or using a calculator, you can find that the z-score separating the bottom 75% from the top 25% is approximately 0.6745.
Now, you can substitute the values into the formula to find x (which corresponds to Q3):
x = 0.6745 * $12,034 + $17,776
Calculating this expression will give you the answer:
x ≈ $25,838.78
Therefore, the Q3 value for the normal distribution with μ = $17,776 and σ = $12,034 is approximately $25,838.78.
Comparing this result to the actual Q3 value of $22,500 from the data indicates that the right-skew of the distribution pushes the observed value of Q3 lower than what would be expected in a normal distribution.