3. Describing Data Numerically
Mean
2:40 minutesProblem 3.1.35a
Textbook Question
Degrees of Freedom Five recent U.S. presidents had a mean age of 56.2 years at the time of their inauguration. Four of these ages are 64, 46, 54, and 47.
a. Find the missing value.
Verified step by step guidance1
Step 1: Recall the formula for the mean: \( \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}} \). In this case, the mean age is given as 56.2 years, and there are 5 values in total.
Step 2: Represent the missing value as \( x \). The sum of all ages can be expressed as \( 64 + 46 + 54 + 47 + x \). Substitute the mean and the number of values into the formula: \( 56.2 = \frac{64 + 46 + 54 + 47 + x}{5} \).
Step 3: Multiply both sides of the equation by 5 to eliminate the denominator: \( 56.2 \times 5 = 64 + 46 + 54 + 47 + x \).
Step 4: Calculate the sum of the known ages: \( 64 + 46 + 54 + 47 \). Subtract this sum from \( 56.2 \times 5 \) to isolate \( x \).
Step 5: Solve for \( x \), which represents the missing age. This will give you the missing value needed to complete the set of ages.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mean
The mean is the average of a set of values, calculated by summing all the values and dividing by the number of values. In this context, the mean age of the five presidents is given as 56.2 years, which serves as a reference point to find the missing age. Understanding how to manipulate the mean is crucial for solving the problem.
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Degrees of Freedom
Degrees of freedom refer to the number of independent values or quantities that can vary in a statistical calculation. In this case, with five presidents, the degrees of freedom are relevant when calculating the mean, as knowing four ages allows us to determine the fifth. This concept is essential for understanding how constraints affect statistical calculations.
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Solving for a Variable
Solving for a variable involves rearranging an equation to isolate the variable of interest. In this scenario, we need to find the missing age by setting up an equation based on the mean. This process requires basic algebraic skills to manipulate the equation correctly and derive the unknown value.
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