# Linear Transformations Worksheet ## Section A: Multiple Choice Questions
Q1: If a data set has a mean of 50 and each value is increased by 10, what is the new mean? (a) 50 (b) 60 (c) 40 (d) 500
Solution:
Ans: (b) Explanation: When a constant is added to each value in a data set, the mean increases by that constant. Since 10 is added to each value, the new mean is \(50 + 10 = 60\).
Q2: A data set has a standard deviation of 8. If each value is multiplied by 3, what is the new standard deviation? (a) 8 (b) 11 (c) 24 (d) 64
Solution:
Ans: (c) Explanation: When each value is multiplied by a constant, the standard deviation is also multiplied by the absolute value of that constant. Therefore, the new standard deviation is \(8 \times 3 = 24\).
Q3: The median of a data set is 25. If 7 is subtracted from each value, what is the new median? (a) 25 (b) 18 (c) 32 (d) 175
Solution:
Ans: (b) Explanation: The median is a measure of center. When a constant is subtracted from each value, the median also decreases by that constant. Thus, the new median is \(25 - 7 = 18\).
Q4: If the variance of a data set is 16 and each value is increased by 5, what is the new variance? (a) 21 (b) 16 (c) 80 (d) 11
Solution:
Ans: (b) Explanation:Variance is a measure of spread. When a constant is added to or subtracted from each value, the variance remains unchanged because the spread of the data does not change. The new variance is still 16.
Q5: The range of a data set is 30. If each value is multiplied by 2, what is the new range? (a) 30 (b) 32 (c) 60 (d) 15
Solution:
Ans: (c) Explanation: The range is the difference between the maximum and minimum values. When each value is multiplied by a positive constant, the range is also multiplied by that constant. Therefore, the new range is \(30 \times 2 = 60\).
Q6: A linear transformation is applied to a data set: \(Y = 4X + 7\). If the original mean is 10, what is the mean of \(Y\)? (a) 10 (b) 40 (c) 47 (d) 17
Solution:
Ans: (c) Explanation: For a linear transformation \(Y = aX + b\), the new mean is calculated as \(\mu_Y = a\mu_X + b\). Here, \(\mu_Y = 4(10) + 7 = 40 + 7 = 47\).
Q7: If the interquartile range (IQR) of a data set is 12 and each value is divided by 4, what is the new IQR? (a) 12 (b) 3 (c) 48 (d) 8
Solution:
Ans: (b) Explanation: The interquartile range is a measure of spread. When each value is divided by a constant, the IQR is also divided by that constant. Therefore, the new IQR is \(12 \div 4 = 3\).
Q8: A data set has a mean of 20 and standard deviation of 5. The transformation \(Z = \frac{X - 20}{5}\) is applied. What is the standard deviation of \(Z\)? (a) 5 (b) 0 (c) 1 (d) 20
Solution:
Ans: (c) Explanation: This transformation is called standardization. When \(Z = \frac{X - \mu}{\sigma}\), the resulting data set has a mean of 0 and a standard deviation of 1.
## Section B: Fill in the Blanks Q9:When a constant is added to each value in a data set, the measures of center change, but the measures of __________ remain unchanged.
Solution:
Ans: spread Explanation:Measures of spread such as variance, standard deviation, range, and IQR do not change when a constant is added because the relative distances between data points remain the same.
Q10:For the linear transformation \(Y = aX + b\), the formula for the new variance is \(\sigma_Y^2 = \) __________.
Solution:
Ans: \(a^2\sigma_X^2\) Explanation: When a linear transformation is applied, the variance is multiplied by the square of the coefficient \(a\). The constant \(b\) does not affect variance.
Q11:If each value in a data set is multiplied by -2, the mean is multiplied by __________ and the standard deviation is multiplied by __________.
Solution:
Ans: -2 and 2 Explanation: The mean is multiplied by -2 (preserving the sign). The standard deviation is multiplied by the absolute value, which is \(|-2| = 2\), since standard deviation is always non-negative.
Q12:A transformation of the form \(Y = aX + b\) where \(a\) and \(b\) are constants is called a __________ transformation.
Solution:
Ans: linear Explanation: This is the definition of a linear transformation, which involves multiplying by a constant and adding a constant.
Q13:If the first quartile \(Q_1\) of a data set is 15 and each value is increased by 8, the new first quartile is __________.
Solution:
Ans: 23 Explanation:Quartiles are measures of position. When a constant is added to each value, all quartiles increase by that constant. Thus, \(Q_1 = 15 + 8 = 23\).
Q14:The process of transforming data to have a mean of 0 and a standard deviation of 1 is called __________.
Solution:
Ans: standardization Explanation:Standardization (or creating z-scores) transforms data using \(Z = \frac{X - \mu}{\sigma}\) to create a distribution with mean 0 and standard deviation 1.
## Section C: Word Problems Q15:A class of students took a math test with a mean score of 72 and a standard deviation of 10. The teacher decides to curve the grades by multiplying each score by 1.1 and then adding 5 points. What are the new mean and standard deviation?
Solution:
Ans: Using the transformation \(Y = 1.1X + 5\): New mean: \(\mu_Y = 1.1(72) + 5 = 79.2 + 5 = 84.2\) New standard deviation: \(\sigma_Y = 1.1(10) = 11\)
Final Answer: The new mean is 84.2 and the new standard deviation is 11.
Q16:The weights of packages at a shipping facility have a median of 25 kg and an IQR of 8 kg. Due to new packaging materials, each package now weighs 3 kg more. What are the new median and IQR?
Solution:
Ans: When a constant is added to each value: New median: \(25 + 3 = 28\) kg New IQR: The IQR remains unchanged at 8 kg because adding a constant does not affect the spread.
Final Answer: The new median is 28 kg and the new IQR is 8 kg.
Q17:Temperature readings in Celsius for a week have a mean of 20°C and a variance of 9°C². Convert these measurements to Fahrenheit using the formula \(F = \frac{9}{5}C + 32\). What are the mean and variance in Fahrenheit?
Solution:
Ans: Using the transformation \(F = \frac{9}{5}C + 32\): New mean: \(\mu_F = \frac{9}{5}(20) + 32 = 36 + 32 = 68°F\) New variance: \(\sigma_F^2 = \left(\frac{9}{5}\right)^2(9) = \frac{81}{25}(9) = \frac{729}{25} = 29.16\) °F²
Final Answer: The mean is 68°F and the variance is 29.16°F².
Q18:A factory produces bolts with lengths that have a mean of 5 cm and a standard deviation of 0.2 cm. Quality control requires standardizing the measurements. If a bolt has a length of 5.4 cm, what is its standardized score (z-score)?
Solution:
Ans: The z-score formula is \(z = \frac{x - \mu}{\sigma}\) \(z = \frac{5.4 - 5}{0.2} = \frac{0.4}{0.2} = 2\)
Final Answer: The z-score is 2.
Q19:The prices of items in a store have a range of $40. During a sale, all prices are reduced by 20% (multiplied by 0.8). What is the new range of prices?
Solution:
Ans: When each value is multiplied by a constant, the range is also multiplied by that constant. New range = \(40 \times 0.8 = 32\) dollars
Final Answer: The new range is $32.
Q20:A data set representing daily sales (in dollars) has a mean of $200 and a standard deviation of $30. The company converts all values to hundreds of dollars by dividing by 100. What are the new mean and standard deviation in hundreds of dollars?
Solution:
Ans: Using the transformation \(Y = \frac{X}{100}\): New mean: \(\mu_Y = \frac{200}{100} = 2\) (hundreds of dollars) New standard deviation: \(\sigma_Y = \frac{30}{100} = 0.3\) (hundreds of dollars)
Final Answer: The new mean is 2 hundred dollars and the new standard deviation is 0.3 hundred dollars.
pdf , Exam, practice quizzes, Worksheet (with Solutions): Linear Transformations, MCQs, Worksheet (with Solutions): Linear Transformations, Objective type Questions, Sample Paper, video lectures, study material, Viva Questions, Free, Important questions, Extra Questions, ppt, mock tests for examination, Worksheet (with Solutions): Linear Transformations, Previous Year Questions with Solutions, shortcuts and tricks, Semester Notes, Summary, past year papers;
