https://file.aiquickdraw.com/mathbot/file/17309041500735h0hguuw.jpg

To solve this problem, let’s break down each part using the information provided.

### Given Data:
- Mean speed (\(\mu\)) = 23.81 mph
- Standard deviation (\(\sigma\)) = 3.59 mph
- We need to find the middle 95% of all car speeds.

### Part (a): Finding the Middle 95% Range Using a Normal Model
The middle 95% of a normal distribution typically lies between \(\mu - 1.96\sigma\) and \(\mu + 1.96\sigma\), as 95% of values fall within approximately 1.96 standard deviations of the mean.

1. Calculate the lower bound:
   \[
   \text{Lower bound} = \mu - 1.96 \times \sigma = 23.81 - (1.96 \times 3.59)
   \]
   \[
   = 23.81 - 7.0364 = 16.77 \, \text{(rounded to two decimal places)}
   \]

2. Calculate the upper bound:
   \[
   \text{Upper bound} = \mu + 1.96 \times \sigma = 23.81 + (1.96 \times 3.59)
   \]
   \[
   = 23.81 + 7.0364 = 30.85 \, \text{(rounded to two decimal places)}
   \]

So, the cutoff points for the middle 95% of all car speeds are **16.77 mph** and **30.85 mph**.

### Part (b): Comparing with Actual Summary Statistics
For part (b), you would need to access the actual summary statistics table provided (or the values in the accompanying dataset) to compare these model predictions to the actual cutoff points observed in the data. Based on the actual values, you would answer whether the model predictions align closely with the actual data or if there are notable differences.

#### Sample Response for Part (b):
- **If the actual cutoff points closely match 16.77 mph and 30.85 mph**, the model does well in predicting these values, and you would likely not be surprised by this outcome.
- **If the actual cutoff points are significantly different from 16.77 mph and 30.85 mph**, the model may not be perfectly accurate for this data, and there might be some surprising elements.

Would you like further clarification or details on any of the steps?

---

### Additional Related Questions:
1. How do we interpret standard deviation in the context of this problem?
2. What would be the middle 99% range for car speeds using this data?
3. How would the cutoff points change if the standard deviation were larger?
4. What does a normal distribution assumption imply about the data in this problem?
5. Why might real-world data deviate from a perfectly normal distribution?

### Tip:
When working with normal distributions, remember that approximately 68% of values lie within 1 standard deviation, 95% within 2, and 99.7% within 3 standard deviations from the mean.

A traffic engineer recorded the speeds of cars driving past her house, where the speed limit read 20 mph. The mean of a sample of 100 readings was 23.81 mph, with a standard deviation of 3.59 mph. Complete parts a and b below.

a) Using a Normal model, what values should border the middle 95% of all car speeds?
The cutoff points for the middle 95% of all car speeds are ___ mph and ___ mph.

b) The accompanying table lists some summary statistics. From your answer in part a, how well does the model do in predicting those percentiles? Are you surprised? Explain.

Learn how to calculate the 95% confidence interval for car speeds in a Normal distribution. Given a mean of 23.81 mph and a standard deviation of 3.59 mph, this problem guides you through finding cutoff points for the middle 95% range and compares it to actual summary statistics. Ideal for college-level statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation