Let's solve the problem step by step based on the image provided:

Problem Breakdown:

The problem provides data on miles per gallon (MPG) for cars with a three-cylinder engine. You are asked to calculate the following:

(a) Compute the z-score corresponding to the individual who obtained 35.8 miles per gallon. Interpret this result.

The z-score is given by the formula:

$z = \frac{x - \mu}{\sigma}$

Where:

• $x = 35.8$ (miles per gallon of the individual)
• $\mu = 38.6$ (the mean, as seen in part (a))
• $\sigma = 3.11$ (the standard deviation)

The z-score for the individual is already calculated as $z = -0.90$.

Interpretation: The z-score indicates that the individual's MPG is 0.90 standard deviations below the mean.

(b) Determine the quartiles.

The quartiles are:

• $Q_1 = 36.85$ mpg
• $Q_2 = 38.55$ mpg (this is also the median)
• $Q_3 = 41.05$ mpg

(c) Compute and interpret the interquartile range (IQR).

The interquartile range (IQR) is the difference between $Q_3$ and $Q_1$:

$IQR = Q_3 - Q_1 = 41.05 - 36.85 = 4.20 \text{ mpg}$

Interpretation: The IQR represents the range of the middle 50% of the data. The MPG of the middle half of the cars lies within a 4.20 mpg range.

(d) Determine the lower and upper fences. Are there any outliers?

To calculate the fences: $\text{Lower fence} = Q_1 - 1.5 \times IQR = 36.85 - 1.5 \times 4.20 = 30.55 \text{ mpg}$ $\text{Upper fence} = Q_3 + 1.5 \times IQR = 41.05 + 1.5 \times 4.20 = 47.35 \text{ mpg}$

Any data point below 30.55 mpg or above 47.35 mpg is considered an outlier.

Looking at the data (in the "MPG Data" box), no values are below 30.55 or above 47.35.

Conclusion: There are no outliers.

Final Answers:

• (a) The z-score is $-0.90$ (0.90 standard deviations below the mean).
• (b) Quartiles: $Q_1 = 36.85$, $Q_2 = 38.55$, $Q_3 = 41.05$.
• (c) Interquartile Range (IQR): $4.20$ mpg.
• (d) No outliers since the data is within the fences (30.55 mpg and 47.35 mpg).

Would you like more details or explanations on any step?

Here are five related questions to consider:

1. How is the standard deviation used in calculating the z-score?
2. What are the advantages of using the IQR to measure variability instead of the range?
3. Can there be multiple outliers, and how does the IQR help identify them?
4. How would the calculations change if the mean and standard deviation were different?
5. What is the significance of quartiles in statistical analysis?

Tip: When interpreting z-scores, negative values mean the data point is below the mean, while positive values indicate it is above the mean.