The image you uploaded contains questions related to linear transformations of quantitative data, including z-scores, conversion between Fahrenheit and Celsius, and data analysis in the context of statistics. Let's go over the questions step by step:

3. Conversion of Temperatures (Fahrenheit to Celsius)

You're asked to convert temperatures from Fahrenheit to Celsius using the equation: $C = 0.556F - 17.778$

• (a) What is the shape of the distribution of temperatures in degrees Celsius? To answer this, you'd note that a linear transformation (like converting from Fahrenheit to Celsius) preserves the shape of the distribution. Therefore, the shape in Celsius is the same as the shape in Fahrenheit, which can be inferred from the given distribution (likely a roughly symmetric or skewed distribution).

• (b) What is the mean and standard deviation of the distribution of temperatures in degrees Celsius? You are given the mean and standard deviation in Fahrenheit:

  • Mean in Fahrenheit = 87
  • Standard deviation in Fahrenheit = 7.227

  The mean and standard deviation transform linearly. Using the equation $C = 0.556F - 17.778$:

  • For the mean: Multiply by 0.556 and subtract 17.778.
  • For the standard deviation: Multiply only by 0.556, since shifts (subtracting constants) don't affect variability.

4. Quarterback Rating (QBR)

You are asked to calculate z-scores and interpret data related to QBR.

• (a) Calculate and interpret the z-score for Patrick Mahomes, who has a QBR of 63. Use the z-score formula: $z = \frac{x - \mu}{\sigma}$ where $x = 63$, the mean $\mu = 115.3$, and the standard deviation $\sigma = 30.06$.

• (b) Describe the mathematical operations used to calculate the z-score. You subtract the mean from Patrick Mahomes' QBR and then divide by the standard deviation.

• (c) Convert all 30 quarterback ratings into z-scores. A linear transformation of all the data into z-scores will result in the distribution having a mean of 0 and a standard deviation of 1.

5. Long Jump Distances for Women

This question focuses on converting data from meters to feet and calculating z-scores.

• (a) Calculate the z-score for Jackie Joyner-Kersee, who jumped 7.49 meters. Using the z-score formula and the provided statistics:

  • Mean $\mu = 7.017$
  • Standard deviation $\sigma = 0.2117$
  • Jump $x = 7.49$

• (b) How far did Jackie Joyner-Kersee jump in feet? Convert meters to feet using the conversion factor $1 \, \text{meter} = 3.28 \, \text{feet}$.

• (c) If all 50 jumps were converted to feet, calculate the new mean and standard deviation. The mean and standard deviation in feet will be calculated by multiplying the values in meters by 3.28.

• (d) Calculate the z-score for her jump in feet. The z-score remains the same under linear transformations, as multiplying by a constant does not affect the z-score.

Would you like me to calculate any specific values or provide more details for any part? Here are five related questions to consider:

1. How does converting temperatures between Fahrenheit and Celsius affect statistical measures like the median or quartiles?
2. What impact does a z-score greater than 0 or less than 0 have on interpreting an athlete's performance?
3. How would the conversion of a dataset to z-scores affect outliers?
4. Why do transformations like converting units (e.g., meters to feet) not affect the shape of the distribution?
5. How is the standard deviation affected by multiplication of all data points by a constant?

Tip: When working with z-scores, a positive z-score means the data point is above the mean, while a negative z-score means it is below the mean.