Question

The mean and standard deviation of \(\mathrm{x}\) is 40 and 4 respectively the mean and standard deviation of \([(x-40) / 4]\) is (a) 1,0 (b) 1,1 (c) 0,1 (d) \(0,-1\)

Short answer

The mean and standard deviation of the transformed variable \([(x-40) / 4]\) are 0 and 1, respectively. Therefore, the correct answer is (c) 0, 1.

Step-by-step solution

Find the mean of the transformed variable

To find the mean of the transformed variable \([(x-40) / 4]\), we will calculate the expected value of this expression. Since the mean of \(x\) is given as 40, we have: \[E[(x-40)/4] = \frac{1}{4}E[(x-40)] = \frac{1}{4}(E[x]-40)\] Now, substitute the given mean of \(x\), 40, into the equation: \[\frac{1}{4}(40-40) = \frac{1}{4}(0) = 0\] So, the mean of the transformed variable is 0.

Find the standard deviation of the transformed variable

To find the standard deviation of the transformed variable \([(x-40) / 4]\), we will first find the variance of this expression. The variance of a constant times a variable is equal to the square of the constant times the variance of the variable: \[\text{Var}[(x-40)/4] = \left(\frac{1}{4}\right)^2\text{Var}[(x-40)]\] Since adding a constant doesn't change the variance, \[\text{Var}[(x-40)/4] = \left(\frac{1}{4}\right)^2\text{Var}[x]\] Now, substitute the given standard deviation of \(x\), 4, into the equation (recall that the variance is the square of the standard deviation): \[\left(\frac{1}{4}\right)^2 (4^2) = (1/16)(16) = 1\] Now, we find the standard deviation by taking the square root of the variance: \[\sqrt{1} = 1\] So, the standard deviation of the transformed variable is 1.

Match the calculated mean and standard deviation with the given options

We found that the mean of the transformed variable is 0 and its standard deviation is 1. By comparing our results with the given options, we see that the answer corresponds to option (c) 0, 1.

Key concepts

Transformed Variables

In statistics, a transformed variable occurs when we modify an original variable to achieve a desired effect on its distribution. This is often done through mathematical operations like scaling or shifting. In this exercise, the transformation applied is \((x-40)/4\).

• **Scaling**: Divides the original value by a constant (in this case, 4), which can reduce variance and standard deviation.

• **Shifting**: Subtracting 40 from each value moves the mean to a new position.

Transformations are useful for simplifying data, stabilizing variances, or meeting certain analysis assumptions. Recognizing how transformations affect variables helps us make accurate statistical inferences.

Expected Value

Expected value, often known as the mathematical expectation, is essentially the average value of a random variable over numerous trials or occurrences. It's a critical concept in probability and statistics.

In our exercise, the expected value of the transformed variable \((x-40)/4\) is calculated to understand its mean. You utilize the fact that the expected value of a linear transformation, such as \((ax+b)\), of a random variable \(x\) is \(aE(x) + b\).

For the transformation in question:

• Subtracting the mean 40 results in a mean shift to zero: \(E[x] - 40 = 0\).
• The expected value simplifies to zero as: \(\frac{1}{4}(0) = 0\).

This demonstrates how transformations can be used methodically to deduce new expected values.

Variance Calculation

Variance measures the spread or dispersion of a set of values. It is a foundational concept for understanding variability in data.

Our exercise shows how to calculate variance for a transformed variable. The rule is that the variance of a transformation \((ax + b)\) of a variable \(x\) is \(a^2 \text{Var}(x)\), as the b (shift) doesn't influence the spread.

Applying this in our problem,

• The transformation \((x-40)/4\) gives us a scaling factor of \(1/4\).
• Thus, the variance becomes \(\left(\frac{1}{4}\right)^2 \text{Var}(x)\).

The exercise uses the given standard deviation, \(\sigma = 4\), hence \(\text{Var}(x) = \sigma^2 = 16\).

Plugging this in, variance simplifies to 1, showing the effect of scaling on variability.

Standard Deviation Transformation

Standard deviation transformation is closely tied to variance. It's the square root of the variance, and provides insights into the data's spread around the mean.

In our example, when each original variable \(x\) is altered using \((x-40)/4\), we consider the impact on standard deviation.

The transformation scaling factor \(1/4\) means that the new standard deviation is:

• \(\text{SD}[(x-40)/4] = \sqrt{\text{Var}[(x-40)/4]} = \sqrt{1} = 1\).

This demonstrates how dividing by a constant reduces the spread, creating a smaller range of values around the new mean. Such transformations are crucial when adjusting datasets for comparability or to match specific analytical criteria.