Question

In City X last April, was the average (arithmetic mean) daily high temperature greater than the median daily high temperature?
(1) In City X last April, the sum of the 30 daily high temperatures was 2,160°.
(2) In City X last April, 60 percent of the daily high temperatures were less than the average daily high temperature.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
• C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient to answer the question asked.
• E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Hint:

Understand the conceptual relationship between mean and median. The median is the 50th percentile mark. If a statement tells you what percentage of the data falls below the mean, you can directly compare the mean to the median. If more than 50% of the data is below the mean, the mean is greater than the median.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question compares the mean and median of a set of data.

Mean (Average): The sum of the values divided by the number of values.
Median: The middle value of a data set when it is sorted. If there is an even number of values, it's the average of the two middle values. The median represents the 50th percentile: 50% of the data is at or below the median.
The question is: Is Mean>Median? This often happens in a right-skewed distribution, where a few high values pull the mean up.
Step 2: Detailed Explanation:
The data set consists of 30 daily high temperatures. Analyze Statement (1): The sum of the 30 daily high temperatures was 2,160°.
We can calculate the mean from this information: \[ \text{Mean} = \frac{\text{Sum}}{\text{Count}} = \frac{2160}{30} = 72^\circ \] This gives us the exact value of the mean. However, we have no information about the individual data points, so we cannot determine the median. The median could be less than, equal to, or greater than 72. Statement (1) is not sufficient.
Analyze Statement (2): 60 percent of the daily high temperatures were less than the average daily high temperature.
Let the 30 temperatures be sorted in increasing order: \(t_1, t_2, ..., t_{30}\). The median is the value that splits the data set in half. For 30 values, 50% of the data (15 values) are less than or equal to the median, and 50% (15 values) are greater than or equal to the median. The median is calculated as \((t_{15} + t_{16}) / 2\). The median is the 50th percentile. The statement says that 60% of the temperatures are less than the mean. This means that the mean is greater than at least 60% of the data points. Since the median is the point where only 50% of the data is smaller, and the mean is a point where 60% of the data is smaller, the mean must be greater than the median. Mean>60th percentile value. Median = 50th percentile value. Therefore, Mean>Median. The answer to the question is definitively "Yes". Statement (2) is sufficient.
Step 3: Final Answer:
Statement (2) alone is sufficient to answer the question.