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Chapter 3: Problem 34

Student loans \((2.2)\) A government report looked at the amount borrowed for college by students who graduated in 2000 and had taken out student loans. 12 The mean amount was \(\overline{x}=\$ 17,776\) and the standard deviation was \(s_{x}=\$ 12,034\) . The median was \(\$ 15,532\) and the quartiles were \(Q_{1}=\$ 9900\) and \(Q_{3}=\$ 22,500\) (a) Compare the mean and the median. Also compare the distances of \(Q_{1}\) and \(Q_{3}\) from the median. Explain why both comparisons suggest that the distribution is right-skewed. (b) 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 \(\mu=\$ 17,776\) and \(\sigma=\$ 12,034\) and compare it with \(Q_{3}=\$ 22,500\) .

Short Answer

Expert verified
The mean is higher than the median and \( Q_3 \) is farther from the median than \( Q_1 \), indicating right-skewness. The normal distribution鈥檚 \( Q_3 \) is \( \$ 25,896 \), larger than the actual \( Q_3 \) of \( \$ 22,500 \).

Step by step solution

01

Mean and Median Comparison

To compare the mean and the median, note that the mean amount of loans borrowed is \( \overline{x} = \\( 17,776 \) and the median is \( \\) 15,532 \). The mean is greater than the median. In a right-skewed distribution, the mean is typically larger than the median because high values excessively influence the mean.
02

Quartile Distance Comparison

Calculate the distance of \( Q_1 \) and \( Q_3 \) from the median. \( Q_1 \) is \( \\( 9,900 \) and \( Q_3 \) is \( \\) 22,500 \). The distance from the median to \( Q_1 \) is \( 15,532 - 9,900 = \\( 5,632 \), while the distance from the median to \( Q_3 \) is \( 22,500 - 15,532 = \\) 6,968 \). Since \( Q_3 \) is farther from the median than \( Q_1 \), this suggests right skewness.
03

Identify Normal Distribution Properties

The problem suggests that a normal distribution with the same mean and standard deviation as the data has a higher \( Q_3 \). For a normal distribution, the third quartile is located at \( \mu + 0.675\sigma \).
04

Calculate Third Quartile for Normal Distribution

Calculate \( Q_3 \) for the normal distribution using \( \mu = \\( 17,776 \) and \( \sigma = \\) 12,034 \). Use the formula: \[ Q_3 = \mu + 0.675 \times \sigma = 17,776 + 0.675 \times 12,034. \]
05

Calculation Result

Calculate: \( 0.675 \times 12,034 = 8,120 \). Thus, \( Q_3 = 17,776 + 8,120 = \$ 25,896 \).
06

Compare Normal Distribution Quartile with Actual Quartile

The third quartile of the normal distribution is \( \\( 25,896 \). This is indeed larger than the actual third quartile \( Q_3 = \\) 22,500 \), confirming the effect of right-skewness where extreme upper values increase the standard deviation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Right-skewed distribution
A right-skewed distribution occurs when most of the data points are clustered towards the lower end of the scale, with a few extreme values stretching towards the higher end. This creates a long tail on the right side of the distribution. In simpler terms, imagine a classroom of students where most students scored around 70 to 80 on a test, but a few scored above 95. These high scores pull the average (mean) up, making it higher than the point at which half the data lies (the median). This scenario is typical for datasets displaying right skewness. Understanding how the extreme values affect the mean is crucial for interpreting data correctly.
Mean and median comparison
Comparing the mean and median is a useful way to determine the skewness of a distribution. The mean is the arithmetic average of all data points. It's calculated by adding all the values and then dividing by the number of values. The median, on the other hand, is the middle point of the data when it's arranged in order. If you have an odd number of observations, it's the exact middle value, and for an even number, it's the average of the two middle values. In the problem, the mean loan amount is $17,776, while the median is $15,532. Since the mean is greater than the median, we can infer that the distribution is right-skewed. In right-skewed distributions, the mean is typically larger because those higher values have a disproportionate effect, pulling the average upwards.
Quartiles
Quartiles are values that divide a dataset into four equal parts. They help describe the spread and shape of a dataset by indicating variability and skewness. - **First Quartile (Q1)**: This is the median of the lower half of the data, meaning that 25% of the data points are less than this value. - **Third Quartile (Q3)**: The median of the upper half of the data, indicating that 75% of data points fall below this value. In the exercise, Q1 is $9,900, and Q3 is $22,500. These numbers not only help define the range of middle 50% of the data (known as the interquartile range, or IQR) but also the skewness. With Q3 being farther from the median ($15,532) than Q1, it shows the data is right-skewed, as the higher end of the data stretches further away from the median.
Standard deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means the data points are close to the mean, while a high standard deviation indicates that the data points are spread out over a broader range of values. In a right-skewed dataset, such as the one described, the standard deviation can be misleadingly high. This happens because the extreme values that contribute to the right tail also increase the spread of the data. Thus, the dataset appears to have more variability than there might be among most data points. For example, with this dataset having a high standard deviation ($12,034), it seems like there's a wide range in student loan amounts, potentially more than what would be typical if the outliers were not as extreme. This attribute of right-skewed data is why it's crucial to consider skewness when interpreting standard deviation for any dataset.

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