91Ó°ÊÓ

According to Statistical Abstract of the United States, 2012, average salary (in dollars) of primary and secondary school classroom teachers in 2009 in the United States varied among states with a fivenumber summary of: minimum \(=35,070, \mathrm{Q} 1=45,840,\) median \(=48,630, \mathrm{Q} 3=55,820,\) maximum \(=69,119\) (Data available in the teacher_salary file.) a. Find and interpret the range and interquartile range. b. Sketch a box plot, marking the five-number summary on it. c. Predict the direction of skew for this distribution. Explain. d. If the distribution, although skewed, is approximately bell shaped, which of the following would be the most realistic value for the standard deviation: (i) 100 , (ii) 1000 , (iii) 7000 , or (iv) 25,000 ? Explain your reasoning.

Step by step solution

01

Finding the Range

The range of a data set is calculated by subtracting the minimum value from the maximum value. Range = Maximum - Minimum = 69,119 - 35,070 = 34,049. So, the range is 34,049, which indicates the spread of the salaries from the lowest to the highest.

02

Calculating the Interquartile Range (IQR)

The interquartile range is calculated by subtracting the first quartile (Q1) from the third quartile (Q3).\[ IQR = Q3 - Q1 = 55,820 - 45,840 = 9,980 \]The IQR is 9,980, which represents the range of the middle 50% of the salaries.

03

Sketching the Box Plot

To sketch a box plot, draw a box from Q1 to Q3 and a line inside the box at the median. - Mark 35,070 at the left for the minimum. - Mark 45,840 at the beginning of the box for Q1. - Draw a line at 48,630 for the median. - Mark 55,820 at the end of the box for Q3. - Mark 69,119 at the right for the maximum. The box plot visually shows the five-number summary of the data.

04

Predicting the Direction of Skew

To predict the skew direction, consider how the whiskers on the box plot are spread. - The range from the minimum to Q1 is smaller than from Q3 to the maximum, suggesting a longer right whisker. - This indicates a right skew (or positive skew), meaning the distribution tails off more to the right.

05

Choosing the Realistic Standard Deviation

If the distribution is approximately bell-shaped, a rough estimate of the standard deviation is a quarter of the range.Estimate: \(\frac{34,049}{4} \approx 8,512\).Considering the options, a standard deviation of 1,000 (option ii) would be the most realistic, as the values 7,000 and 25,000 are too high and 100 is too low.

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Key Concepts

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

Interquartile Range (IQR)

The interquartile range, or IQR, is a useful measure of variability in a data set. It focuses on the middle 50% of the data, thus giving a clear picture of where the bulk of the values lie. To find the IQR, you subtract the first quartile (Q1) from the third quartile (Q3).For our teacher salary example, this would be:\[ IQR = Q3 - Q1 = 55,820 - 45,840 = 9,980 \]This number, 9,980, shows the range within which the central half of the teacher salaries fall. This is particularly helpful because it excludes outliers or extreme values that might skew the analysis. Unlike the regular range, which only provides a view on the spread of all data points, the IQR focuses on the spread within which most data points lie, offering a more robust understanding.

Box Plot

A box plot, also known as a whisker plot, is a graphical representation of a data set’s distribution through its five-number summary: the minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Here's how to create one: - Draw a horizontal or vertical line, and mark the minimum (35,070) and maximum (69,119) at the ends. - Create a box from Q1 (45,840) to Q3 (55,820). - Inside the box, draw a line where the median (48,630) is located. The box itself represents the IQR, and the lines extending from it, known as "whiskers," exhibit the rest of the data’s spread. This visual helps us quickly gauge the concentration of data and identify potential outliers. The symmetry or asymmetry of the box and whiskers can also suggest the data's skewness.

Data Skewness

Data skewness describes the asymmetry, or lack thereof, of a distribution. If we imagine folding a graph of the data at the median, skewness tells us if one side is longer or fatter than the other. This can occur even in data sets that might appear relatively symmetrical at first glance. For teacher salaries, we note that the right whisker extends further than the left one in our box plot. This indicates a right skew, or positive skewness, suggesting that there are a few salaries significantly higher than the rest, pulling the mean to the right. Recognizing skewness helps in understanding data distribution and selecting the appropriate measures of central tendency.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. Unlike the range, which focuses on the distance between the extreme values, standard deviation gives insight into how much the individual data points typically differ from the mean.When a distribution is approximately bell-shaped, a useful rule of thumb is that most data points (about 68%) lie within one standard deviation of the mean.In the problem, we estimated the standard deviation to be roughly a quarter of the range:\[ \frac{34,049}{4} \approx 8,512 \]Our estimate helps in deciding that a standard deviation of 1,000 is the most realistic among the given options. This shows the dispersion level that accommodates the bulk of the data while considering the skewed nature of the salaries.