Problem 21 - Algebra 2 and Trigonometry

Question

Each time Mrs. Taggart fills the tank of her car, she estimates, from the number of miles driven and the number of gallons of gasoline needed to fill the tank, the fuel efficiency of her car, that is, the number of miles per gallon. The table shows the result of the last 20 times that she filled the car. a. Find the mean and the median fuel efficiency (miles per gallon) for her car. b. Find the percentile rank of 34 miles per gallon. $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline \text { Miles per } & {32} & {33} & {34} & {35} & {36} & {37} & {38} & {39} & {40} \\ \hline \text { Gallon } & {} & {} & {1} & {3} & {2} & {5} & {3} & {3} & {2} & {0} & {1} \\\ \hline\end{array} $$

Step-by-Step Solution

Answer

Mean: 33.95 mpg, Median: 37 mpg, 34 mpg is at the 2.5 percentile rank.

Step 1: Interpret the Table

The table represents how many times each miles per gallon (mpg) value was observed. For example, 34 mpg was observed once, 35 mpg three times, and so on.

Step 2: Calculate the Mean

Calculate the mean by multiplying each mpg value by the number of times it was observed, summing up the results, and dividing by the total number of observations (20).Calculate:\[\text{Mean} = \frac{32(0) + 33(0) + 34(1) + 35(3) + 36(2) + 37(5) + 38(3) + 39(3) + 40(2) + 41(0)X}{20}\]After calculating the above:\[\text{Mean} = \frac{679}{20} = 33.95\]

Step 3: Arrange the Data for Median Calculation

Arrange the data in ascending order according to the frequency. The ordered data is:\[34, 35, 35, 35, 36, 36, 37, 37, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40\]

Step 4: Calculate the Median

The median is the middle value. Since there are 20 observations, the median is the average of the 10th and 11th values. Both the 10th and 11th values are 37, so:\[\text{Median} = 37\]

Step 5: Calculate Percentile Rank for 34 mpg

First, count the number of values less than 34 mpg, which is zero. The number of values equal to 34 mpg is 1.Use the formula for percentile rank:\[\text{Percentile Rank} = \frac{\text{Number of values less than 34} + 0.5(\text{Number of values equal to 34})}{\text{Total number of observations}} \times 100\]Substitute the values:\[\text{Percentile Rank} = \frac{0 + 0.5(1)}{20} \times 100 = 2.5\]

Key Concepts

Mean Calculation

Calculating the mean is about finding the average value from a set of numbers. In Mrs. Taggart's case, it involves finding the average fuel efficiency of her car over 20 fill-ups.
Think of the mean as a balance point in your data, where each individual value contributes proportionally to this balance.
To calculate mean fuel efficiency, follow these steps:

• Multiply each miles per gallon (mpg) value by the number of times it was recorded. For instance, if 35 mpg was observed three times, multiply 35 by 3.
• Add together all these products to get a "total miles." For example, summing 34(1) + 35(3) + 36(2) + 37(5) + 38(3) + 39(3) + 40(2) gives 679.
• Divide this sum by the total number of observations. In this case, 679 divided by 20 gives a mean of 33.95 mpg.

As you can see, the resulting mean gives a simple snapshot of overall car fuel efficiency. But remember, if there are outliers in your observations, they might skew this average. So, it's always good to consider median and mode as well when analyzing the data.

Median Calculation

The median gives you the middle value in a set of numbers. It's like looking at the heart or midpoint of your data. This is helpful because it is less affected by outliers compared to the mean, giving a truer sense of our dataset's center.
In a sorted list of observations for Mrs. Taggart's car, the fuel efficiencies are ordered from lowest to highest. This makes it easy to locate the center.

• First, arrange the mpg values in order. For instance, her data organized from least to greatest is: 34, 35, 35, 35, 36, 36, 37, 37, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40.
• Because she has 20 observations, the median will be the average of the 10th and 11th values in this sequence. Here, both these values are 37.
• Therefore, the median fuel efficiency is 37 mpg.

The median can provide insights that are more representative of how Mrs. Taggart's car performs on a typical basis, especially when her data includes very low or high efficiencies that might skew the mean.

Percentile Rank

The percentile rank shows how one data point compares to the rest in your dataset. In this context, it tells us how Mrs. Taggart’s fuel efficiency of 34 mpg ranks relative to other recorded efficiencies.
It's helpful to understand where a single observation stands, especially in situations where comparing relative performance or distribution is important. Here's how you calculate the percentile rank for 34 mpg:

• Count how many mpg values are less than 34. In this set, none are less than 34, so this count is zero.
• Determine how many times 34 mpg appears. Here, it's recorded once.
• Apply the formula: Percentile Rank = [(Number of values less than 34) + 0.5 × (Number of values equal to 34)] / Total number of observations × 100.
• Substitute into the formula: [0 + 0.5(1)] / 20 × 100.
• Performing the arithmetic gives a percentile rank of 2.5.

This indicates that 34 mpg is better than 2.5% of the data. Such a low rank suggests it's closer to the lower efficiencies Mrs. Taggart has recorded.