Problem 2 - AP Psychology Premium Prep 2021

Question

The height of six trees is measured. Is plant 6 taller than the median for all six trees? $$ \begin{array}{|c|c|} \hline \text { Plant } & \begin{array}{c} \text { Height } \\ \text { (inches) } \end{array} \\ \hline 1 & 67 \\ \hline 2 & 61 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|} \hline 3 & 72 \\ \hline 4 & 71 \\ \hline 5 & 66 \\ \hline 6 & 68 \\ \hline \end{array} $$ (A) Yes, the median is 67.3 . (B) No, the median is 67.3 . (C) Yes, the median is 67.5 . (D) No, the median is 67.5 .

Step-by-Step Solution

Answer

Yes, the median is 67.5.

Step 1: Arrange tree heights in order

First, we need to arrange the heights of the six trees in ascending order. The given heights are: 67, 61, 72, 71, 66, and 68. The sorted heights are: 61, 66, 67, 68, 71, and 72.

Step 2: Calculate the median height

Since we have an even number of trees (6), the median will be the average of the middle two heights. The middle heights are 67 and 68. To find the average, add them together and divide by 2: \(\frac{67 + 68}{2} = 67.5\) So the median height is 67.5 inches.

Step 3: Compare Plant 6's height to the median height

Now, we know Plant 6's height is 68 inches and the median height is 67.5 inches. Since 68 is greater than 67.5, Plant 6 is taller than the median height of all six trees. Based on our calculation, the correct answer is: (C) Yes, the median is 67.5.

Key Concepts

Understanding Median Calculation

Calculating the median is an important skill in statistics, especially when analyzing data sets. The median is the middle value in a data set when it's arranged in ascending order. It's a great way to understand the central tendency of the data. Here's how to find it:

• First, arrange the data set from the smallest to the largest value.
• If you have an odd number of values, the median is the middle number.
• For an even number of values, like in our exercise, the median is the average of the two middle numbers.

In this case, for our set of tree heights: 61, 66, 67, 68, 71, and 72, the two middle numbers are 67 and 68. By averaging these, you find the median: \( \frac{67 + 68}{2} = 67.5 \).
Knowing how to calculate the median helps in summarizing and making sense of the data. It also helps in distinguishing the spread and range of values in a data set.

Data Analysis for Tree Heights

Data analysis involves interpreting data to make decisions. In the context of the exercise, you have tree heights that need analysis to determine certain facts—like whether a specific plant is taller than the median height.

• Sort your data: Arrange numbers in order to easily identify median and trends.
• Look at distribution: Check how data varies around the median.
• Identify key values: Determine if an element is above or below significant data points like the median.

In this exercise, we determined that Plant 6 is 68 inches tall. By comparing this with the median calculated earlier (67.5 inches), we found that Plant 6 is indeed taller than the median.
Analyzing data in this way can answer practical questions about real-world scenarios and help make informed predictions.

Step-by-Step Solution Process

Breaking problems into steps simplifies complex tasks into manageable parts. Here's a step-by-step approach, as we saw in the exercise to solve the median-related question:

• Step 1: Arrange heights in order. Start by ordering the heights from smallest to largest: 61, 66, 67, 68, 71, and 72.
• Step 2: Calculate the median. With an even number of trees, find the average of the two middle values, which are 67 and 68. This gives us a median of \( \frac{67 + 68}{2} = 67.5 \).
• Step 3: Compare to Plant 6. Plant 6 is 68 inches tall. Since 68 is greater than 67.5, the answer to the question is "Yes, Plant 6 is taller than the median."

By following this structured approach, you can systematically work through similar problems and ensure accuracy in your solutions. This method not only aids in academic exercises but is also applicable in numerous real-life analytical situations.