Question 42·Hard·One-Variable Data Distributions; Measures of Center and Spread
Data set P consists of 32 observations and has a mean of 45. Data set Q consists of 48 observations. The two data sets are combined, and then 7 is added to every one of the resulting 80 values. The mean of this transformed data set is 70.
What is the mean of the original data set Q?
(Express the answer as an integer)
For problems about combining data sets and shifting all values, first use the transformation rule: adding (or subtracting) a constant to every value changes the mean by that constant. Undo any given shift to find the original combined mean. Then treat the combined mean as a weighted average: total sum is (number of values) × (mean), so express the total sums of each data set in terms of their means, add them, divide by the total number of observations, and set this equal to the known combined mean. Solve the resulting linear equation carefully, watching your arithmetic with products like 32·45 and 63·80 to avoid simple mistakes. © Аnікo
Hints
Think about the effect of adding 7
What happens to the mean of a data set if you add the same number, like 7, to every value? How can you work backward from the new mean of 70?
Find the original combined mean
The mean of 70 is after adding 7 to each of the 80 values. What was the mean of those 80 values before the 7 was added?
Use a weighted average for the combined data
Let x be the mean of Q. Write an equation for the combined mean of the 32 values from P (mean 45) and the 48 values from Q (mean x), and set that equal to the original combined mean you found in the previous hint.
Solve your equation carefully
Once you have an equation with x, clear the fraction, simplify step by step, and solve for x. That value is the mean of Q.
Desmos Guide
Represent the combined mean before adding 7
Let x be the mean of data set Q. In Desmos, enter the expression:
y = (32*45 + 48*x)/80
This represents the original combined mean of all 80 values in terms of x.
Set the combined mean equal to the known value
Add a second line in Desmos:
y = 63
This is the original combined mean found by undoing the addition of 7 (since 70 − 7 = 63).
Find the x-value where the graphs intersect
Zoom or adjust the window until you see where the two lines intersect. The x-coordinate of the intersection point is the value of x that makes the equation true; that x is the mean of data set Q. Аniko AІ Тutоr
Step-by-step Explanation
Undo the effect of adding 7
If you add the same number to every value in a data set, the mean increases by that number. Here, 7 was added to every value and the new mean is 70.
So the mean of the combined data set before adding 7 must be:
That 63 is the mean of all 80 original values (32 from P and 48 from Q).
Express the combined mean in terms of Q's mean
Let be the mean of the original data set Q.
- Data set P: 32 observations, mean 45, so its total sum is .
- Data set Q: 48 observations, mean , so its total sum is .
The combined mean of all 80 original values is 63, so:
Solve the equation for x
Now solve (Аniко.ai)
Multiply both sides by 80:
Compute the products:
Subtract 1440 from both sides:
Divide by 48:
So the mean of the original data set Q is 75.