Question

exercises

1. the following automobile prices are listed in descending order: (a, b, c, d, x, y,) and (w). express the difference between the median and the mean of these prices algebraically.
2. a local charity wants to purchase a classic 1956 thunderbird for use as a prize in a fundraiser. they find the following eight prices in the paper.

($48,000) ($57,000) ($31,000) ($58,999)
($61,200) ($59,000) ($97,500) ($42,500)
a. what is the best measure of central tendency to use to get a reasonable estimate for the cost of the car? explain. find the mean, median, and mode.
b. what is the range?

1. given is the list of prices for a set of used original hubcaps for a 1957 chevrolet. they vary depending on the condition. find the following statistics for the hubcap prices.

($120) ($50) ($320) ($220) ($310) ($100) ($260) ($300) ($155) ($125)
($600) ($250) ($200) ($200) ($125)
a. mean, to the nearest dollar
b. median
c. mode
d. four quartiles
e. range
f. interquartile range

Explanation:

Problem 1 (First Exercise)

Step1: Find the median

There are 7 values (a, b, c, d, x, y, w) in descending order. The median of a set with \( n = 7 \) (odd number of elements) is the middle value, which is the 4th value, so the median is \( d \).

Step2: Find the mean

The mean (\(\bar{x}\)) of a set of numbers is the sum of the numbers divided by the count of numbers. So the mean is \( \frac{a + b + c + d + x + y + w}{7} \).

Step3: Find the difference

The difference between the median and the mean is \( d - \frac{a + b + c + d + x + y + w}{7} \). We can simplify this:
\[

$$\begin{align*} d - \frac{a + b + c + d + x + y + w}{7}&=\frac{7d - (a + b + c + d + x + y + w)}{7}\\ &=\frac{6d - a - b - c - x - y - w}{7} \end{align*}$$

\]

(for Sub - part a):

• Mean: Sum of prices (\(31000 + 42500+\cdots+97500 = 455199\)) divided by 8, gives \( \approx\$56,899.88 \).
• Median: Average of 4th (\( \$57,000 \)) and 5th (\( \$58,999 \)) values, gives \( \approx\$57,999.5 \).
• Mode: No value repeats, so no mode (or all values have frequency 1).
• Best Measure: Median, as the mean is affected by the outlier (\( \$97,500 \)).

The range is the difference between the maximum and minimum values. The maximum value is \( \$97,500 \) and the minimum value is \( \$31,000 \).

Step1: Identify max and min

Max \( = 97500 \), Min \( = 31000 \)

Step2: Calculate range

Range \(=97500 - 31000=66500\)

Answer:

\(\frac{6d - a - b - c - x - y - w}{7}\) (or \(d - \frac{a + b + c + d + x + y + w}{7}\))

Problem 2 (Sub - part a)

First, we need to sort the eight prices: \( \$31,000 \), \( \$42,500 \), \( \$48,000 \), \( \$57,000 \), \( \$58,999 \), \( \$59,000 \), \( \$61,200 \), \( \$97,500 \)

Step1: Find the mean

Sum of the prices: \(31000 + 42500 + 48000 + 57000 + 58999 + 59000 + 61200 + 97500\)
\[

$$\begin{align*} &31000+42500 = 73500\\ &73500 + 48000=121500\\ &121500+57000 = 178500\\ &178500+58999 = 237499\\ &237499+59000 = 296499\\ &296499+61200 = 357699\\ &357699+97500 = 455199 \end{align*}$$

\]
Mean \(=\frac{455199}{8}=56899.875\approx\$56,899.88\)

Step2: Find the median

Since there are 8 values (even number of elements), the median is the average of the 4th and 5th values. The 4th value is \( \$57,000 \) and the 5th value is \( \$58,999 \).
Median \(=\frac{57000 + 58999}{2}=\frac{115999}{2} = 57999.5\approx\$57,999.5\)

Step3: Find the mode

The mode is the value that appears most frequently. In this set, all values appear only once, so there is no mode (or we can say all values are modes with frequency 1).

Step4: Determine the best measure

The data has an outlier (\( \$97,500 \)) which will affect the mean. The median is less affected by outliers. So the best measure of central tendency is the median.