Analysis showed that the mean arrival rate for vehicles at a certain Chevron station on Friday afternoon last year was 4.5 vehicles per minute. How large a sample would be needed to estimate this year's mean arrival rate with 98 percent confidence and an error of ± 0.5? (Round the standard deviation answer to 4 decimal places. Enter your answer as a whole number. Use a z-value taken to four decimal places in your calculations.)

To solve this problem, we need to determine the sample size required to estimate this year's mean arrival rate with a 98% confidence level and an error margin of ±0.5 vehicles per minute.

### Step 1: Identify the given information
- **Mean arrival rate** (\(\mu\)): 4.5 vehicles per minute
- **Confidence level**: 98%
- **Margin of error** (\(E\)): ±0.5 vehicles per minute
- **z-value**: This corresponds to a 98% confidence level.
- **Population standard deviation** (\(\sigma\)): We will assume that last year's standard deviation can be used, and it is estimated as \(\sigma = 4.5\) (as an initial assumption for simplicity since the problem does not explicitly state it).

### Step 2: Find the z-value
For a 98% confidence level, the z-value corresponds to the area in the tails being 1 - 0.98 = 0.02. Each tail will have an area of 0.01. Using the z-table or standard normal distribution, the z-value for a cumulative area of 0.99 is:

\[ z = 2.3263 \]

### Step 3: Calculate the required sample size
The formula for the sample size \(n\) is:

\[
n = \left( \frac{z \cdot \sigma}{E} \right)^2
\]

Plugging in the values:

\[
n = \left( \frac{2.3263 \times 4.5}{0.5} \right)^2
\]

### Step 4: Perform the calculations

\[
n = \left( \frac{10.46835}{0.5} \right)^2 = \left( 20.9367 \right)^2 = 438.3713
\]

### Step 5: Round the result
Since sample size must be a whole number, we round up:

\[
n = 439
\]

### Final Answer:
You would need a sample size of **439** vehicles to estimate this year's mean arrival rate with 98% confidence and an error of ±0.5 vehicles per minute.

Would you like further details or have any questions? 

Here are 5 related questions you might find helpful:

1. How is the z-value determined for different confidence levels?
2. What is the impact of a larger margin of error on the required sample size?
3. How does the population standard deviation affect the sample size calculation?
4. What is the relationship between confidence level and sample size?
5. How would the calculation change if the population mean were known instead of the sample mean?

**Tip:** Always remember to round up the sample size to ensure that your margin of error is within the desired limit.

Learn how to calculate the sample size needed to estimate this year's mean arrival rate with a 98% confidence level and a ±0.5 vehicles per minute margin of error. Detailed step-by-step solution using statistical formulas and concepts. Suitable for advanced undergraduate students and professionals.

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