Question 4·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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The frequency table summarizes the number of training sessions attended by each employee in a department. The value of $k$ is a positive integer, and the mean number of sessions in the original data set is $4.1$ .

Number of sessions	Frequency
$1$	$3$
$2$	$5$
$4$	$k$
$7$	$4$
$9$	$2$

In a corrected version of the data set, each record of $1$ session is changed to $3$ sessions, and each record of $9$ sessions is changed to $5$ sessions. Which choice gives the median and mean of the corrected data set?

A frequency table tells you how many times each value occurs, so a missing frequency affects both the total and the count. Use the given mean to write an equation, then find the missing frequency with a Desmos regression. After the corrections, count into the middle for the median and recalculate the mean. The mean follows the total; the median follows positions.

Hints

Hint 1
The mean is the total number of sessions divided by the number of employees. Each row contributes its session count times its frequency to the total. How can you use the given mean to find $k$ ?
Hint 2
A frequency tells you how many records hold a value. A correction replaces records; it doesn't add employees. Move the affected records to their new session counts, but keep the other rows.
Hint 3
The median is the middle of the values in numerical order. With an even number of employees, check both middle positions. Add the corrected frequencies from the smallest session count upward to find what fills them.

Step-by-step

Find the missing frequency, then update the table
Step 1
Turn the original mean into an equation
Each row contributes its number of sessions times its frequency, or number of employees in that row. The original mean is $4.1$ , so the total sessions must equal $4.1$ times the total number of employees:
$1(3)+2(5)+4k+7(4)+9(2)=4.1(3+5+k+4+2).$
Step 2
Find the missing frequency
Type the equation in Desmos with $\sim$ in place of $=$ . The regression finds the value of $k$ that makes the two sides match. Desmos shows $k=16$ under PARAMETERS, so $16$ employees attended $4$ sessions.
Step 3
Enter the corrected records
The three $1$ s become three $3$ s, and the two $9$ s become two $5$ s. The $2$ s, $4$ s, and $7$ s stay put. Enter the corrected session counts in Desmos column $x_1$ and their frequencies in $y_1$ . Desmos shows each count beside the number of employees who have it.
Step 4
Count the employees
Type $\operatorname{total}(y_1)$ to add the corrected frequencies. Desmos shows $30$ employees, not five: the five table rows represent repeated records. Corrections change values, not the number of employees.
Step 5
Locate the middle records
With $30$ records, the median is the average of the 15th and 16th values in order. The five $2$ s fill positions $1$ through $5$ ; the three $3$ s fill $6$ through $8$ . The sixteen $4$ s then fill $9$ through $24$ . Both middle values are $4$ , so the median is $4$ sessions.
Step 6
Calculate the corrected mean
Type $\frac{\operatorname{total}(x_1y_1)}{\operatorname{total}(y_1)}$ . This multiplies each corrected session count by its frequency, then divides by all $30$ employees. Desmos shows $4.0333\ldots$ ; its fraction button gives $\frac{121}{30}$ . The corrected median is $4$ sessions, and the corrected mean is $\frac{121}{30}$ sessions. Choice D.

Question 4·50 Super-Hard SAT Problem Solving and Data Analysis Questions·Problem Solving and Data Analysis

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