Let's start by organizing the data given, which is:

Original measurements:
2, 3, 1, 0, 42, 3, 1, 0, 4

Part b: Add 3 to each measurement

To obtain the new set of measurements, we add 3 to each of the original values:

$\text{New measurements} = 2+3, 3+3, 1+3, 0+3, 42+3, 3+3, 1+3, 0+3, 4+3$

$\text{New measurements} = 5, 6, 4, 3, 45, 6, 4, 3, 7$

Part 5: Range Calculation

The range is defined as the difference between the maximum and minimum values in the dataset.

For the new measurements:

• Maximum value: 45
• Minimum value: 3

So, the range is:

$\text{Range} = 45 - 3 = 42$

Range = 42

Part 6: Variance (s²) Calculation

The formula for variance $s^2$ is:

$s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1}$

Where:

• $x_i$ are the individual measurements
• $\bar{x}$ is the mean of the measurements
• $n$ is the number of measurements

Step 1: Find the mean $\bar{x}$

The mean $\bar{x}$ is:

$\bar{x} = \frac{5 + 6 + 4 + 3 + 45 + 6 + 4 + 3 + 7}{9} = \frac{83}{9} \approx 9.22$

Step 2: Compute the squared deviations from the mean

For each value $x_i$, we subtract the mean and square the result:

• $(5 - 9.22)^2 = (-4.22)^2 \approx 17.78$
• $(6 - 9.22)^2 = (-3.22)^2 \approx 10.37$
• $(4 - 9.22)^2 = (-5.22)^2 \approx 27.23$
• $(3 - 9.22)^2 = (-6.22)^2 \approx 38.73$
• $(45 - 9.22)^2 = (35.78)^2 \approx 1280.27$
• $(6 - 9.22)^2 = (-3.22)^2 \approx 10.37$
• $(4 - 9.22)^2 = (-5.22)^2 \approx 27.23$
• $(3 - 9.22)^2 = (-6.22)^2 \approx 38.73$
• $(7 - 9.22)^2 = (-2.22)^2 \approx 4.93$

Step 3: Sum the squared deviations

$\sum (x_i - \bar{x})^2 = 17.78 + 10.37 + 27.23 + 38.73 + 1280.27 + 10.37 + 27.23 + 38.73 + 4.93 = 1414.64$

Step 4: Calculate the variance

The variance is:

$s^2 = \frac{1414.64}{9 - 1} = \frac{1414.64}{8} \approx 176.83$

Variance $s^2 \approx 176.8$ (rounded to one decimal place)

Part 6: Standard Deviation (s)

The standard deviation $s$ is the square root of the variance:

$s = \sqrt{176.83} \approx 13.31$

Standard deviation $s \approx 13.31$ (rounded to two decimal places)

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Final Answers:

1. Range = 42
2. Variance $s^2$ = 176.8
3. Standard deviation $s$ = 13.31

Would you like more detailed steps or clarifications on any part of the calculations? Here are some follow-up questions for further exploration:

1. How do I interpret the range in terms of data spread?
2. What effect would adding or subtracting values to the entire dataset have on the variance and standard deviation?
3. How would these results change if the sample size were larger?
4. Can we use this data to find other measures of dispersion, like the interquartile range (IQR)?
5. How does the presence of an outlier (like 42) impact these statistics?

Tip: When calculating the variance, ensure that you subtract the mean from each data point before squaring it; this helps measure how far each value is from the average.