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Let's analyze and solve this statistics question.

### Given Information:
- Mean speed of cars: 28.68 mph
- Standard deviation: 3.38 mph
- Speed limit: 25 mph

The question has two parts:

1. **Part (a)**: How many standard deviations from the mean would a car going the speed limit (25 mph) be?

   To find how many standard deviations a specific value is from the mean, we use the **z-score** formula:
   \[
   z = \frac{X - \mu}{\sigma}
   \]
   where:
   - \( X \) is the value (25 mph here),
   - \( \mu \) is the mean (28.68 mph), and
   - \( \sigma \) is the standard deviation (3.38 mph).

   Substituting the values:
   \[
   z = \frac{25 - 28.68}{3.38} = \frac{-3.68}{3.38} \approx -1.09
   \]

   So, a car traveling at the speed limit (25 mph) is approximately **1.09 standard deviations below the mean**.

2. **Part (b)**: Determine which is more unusual—a car traveling at 40 mph or one traveling at 14 mph.

   We calculate the z-scores for both speeds:
   
   - **For 40 mph**:
     \[
     z = \frac{40 - 28.68}{3.38} = \frac{11.32}{3.38} \approx 3.35
     \]
   - **For 14 mph**:
     \[
     z = \frac{14 - 28.68}{3.38} = \frac{-14.68}{3.38} \approx -4.34
     \]

   Interpretation:
   - A car traveling at 40 mph is approximately **3.35 standard deviations above the mean**.
   - A car traveling at 14 mph is approximately **4.34 standard deviations below the mean**.

   **Conclusion**: The car traveling at 14 mph is more unusual than the car traveling at 40 mph, as it is farther from the mean in terms of standard deviations.

### Answering the Questions
- **(a)** A car traveling at the speed limit (25 mph) is approximately **1.09** standard deviations from the mean.
- **(b)** The correct answer is **B**: "The car traveling 14 mph is more unusual. It is 4.34 standard deviations from the mean, while the car traveling 40 mph is 3.35 standard deviations from the mean."

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Do you have any questions about this solution or want further details?

### Related Questions to Explore:
1. How does the concept of z-scores help in understanding unusual events?
2. What other factors might affect the interpretation of unusual speeds?
3. How would the z-scores change if the standard deviation were larger?
4. What does it mean for a value to be more than 3 standard deviations from the mean in a normal distribution?
5. How would you interpret the mean speed in this context?

### Tip:
When dealing with z-scores, remember that values further from zero (positive or negative) indicate that the value is more unusual or extreme compared to the mean.

John Beale of Stanford, California, recorded the speeds of cars driving past his house, where the speed limit read 25 mph. The mean of 100 representative readings was 28.68 mph, with a standard deviation of 3.38 mph.

(a) How many standard deviations from the mean would a car going the speed limit be?
(b) Which would be more unusual, a car traveling 40 mph or one going 14 mph?

(a) A car traveling at the speed limit is ___ standard deviations from the mean.

(b) Choose the correct answer below:
A. The car traveling 40 mph is more unusual. It is ___ standard deviations from the mean, while the car traveling 14 mph is ___.
B. The car traveling 14 mph is more unusual. It is ___ standard deviations from the mean, while the car traveling 40 mph is ___.
C. Both cars are equally unusual. Both cars are ___ standard deviations from the mean.

Learn how to calculate the Z-score for a car speed of 25 mph based on a mean of 28.68 mph and a standard deviation of 3.38 mph. Determine which car speed, 40 mph or 14 mph, is more unusual by comparing their standard deviations from the mean. Ideal for college-level statistics students.

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