Let's go through the questions one by one.

Question B:

The distance, in miles, traveled to work for each person in a sample is recorded. After the data are collected, all the distances are converted to kilometers by multiplying each value by 1.609. Which of the following statistics will remain the same for both units, miles, and kilometers?

Options:

• A) The median of the distances
• B) The z-scores of the distances
• C) The mean of the distances
• D) The range of the distances
• E) The minimum of the distances

Analysis:

When all distances are multiplied by 1.609 (a constant), some statistics will change, and others will remain the same:

• Mean: It will change because the average distance in kilometers will be 1.609 times the mean distance in miles.
• Median: Similar to the mean, it will also change by the same factor.
• Range: The difference between the maximum and minimum values will increase by 1.609 times.
• Minimum: The smallest value will change because it is multiplied by 1.609.

However, z-scores standardize the data relative to the mean and standard deviation. Since each distance is multiplied by the same constant, the overall shape and relative position of each data point remain the same, so z-scores remain unchanged.

Answer: B) The z-scores of the distances

Question 9:

A local business reports that the standard deviation of all the paychecks from their weekly payroll is $153.47. The owner of the business plans to give all employees a 12% raise along with a one-time $200 bonus. To determine each employee's next paycheck, the following equation is used:

$\text{Next paycheck} = 1.12 (\text{Original paycheck}) + 200$

Which of the following represents the standard deviation of the next paychecks?

Options:

• A) 1.12(153.47) + 200
• B) 1.12(153.47)
• C) 1.12(153.47)^2
• D) 1.12²(153.47)
• E) 1.12²(153.47)²

Analysis:

The key here is understanding how standard deviation behaves under linear transformations:

1. Multiplying by a constant: The standard deviation is scaled by the absolute value of that constant. So, multiplying all paychecks by 1.12 means that the new standard deviation will be 1.12 times the original standard deviation.
2. Adding a constant: Adding or subtracting a constant (the $200 bonus in this case) does not affect the standard deviation because it shifts all data points equally.

Thus, the standard deviation of the next paychecks is simply $1.12 \times 153.47$, as the $200 bonus doesn't affect the spread (standard deviation).

Answer: B) 1.12(153.47)

Would you like more details or have any questions?

Related questions:

1. Why does adding a constant not affect the standard deviation?
2. How do other transformations, such as squaring values, impact standard deviation?
3. What happens to the mean when converting miles to kilometers in Question B?
4. Why are z-scores dimensionless, and how does that help in comparisons?
5. What is the difference between variance and standard deviation when transformations are applied?

Tip:

Remember, standard deviation only changes when data is scaled, not when a constant is added to all values!