A neighborhood group has complained that at an intersection near a school, half of all cars roll through the stop sign without coming to a complete halt. A statistics practitioner plans to observe a random sample of 200 cars. If the neighborhood group is correct, what is the probability that between 45% and 55% of the cars roll through the stop sign? (Round your answer to four decimal places.)

This problem involves the use of the **binomial distribution** and the **normal approximation** to the binomial distribution, because we're working with a large sample size (\(n = 200\)).

Let's break this down step-by-step:

### Step 1: Define parameters of the binomial distribution
- The probability of a car rolling through the stop sign, \(p\), is 0.5 (since half of the cars are claimed to roll through).
- The sample size, \(n\), is 200.
- The binomial distribution can be approximated by a normal distribution when \(n\) is large and \(p\) is not too close to 0 or 1. In this case, it satisfies the conditions for using the normal approximation.

### Step 2: Use normal approximation to the binomial distribution
For the binomial distribution:
- The **mean** \( \mu \) is given by:
  \[
  \mu = np = 200 \times 0.5 = 100
  \]
- The **standard deviation** \( \sigma \) is given by:
  \[
  \sigma = \sqrt{np(1-p)} = \sqrt{200 \times 0.5 \times (1-0.5)} = \sqrt{50} \approx 7.0711
  \]

### Step 3: Apply continuity correction
We want to find the probability that **between 45% and 55%** of the cars roll through the stop sign, i.e., between 90 cars and 110 cars (since \( 0.45 \times 200 = 90 \) and \( 0.55 \times 200 = 110 \)).

To improve the approximation, we apply a continuity correction, adjusting the boundaries by 0.5:
- Lower bound: \( 90 - 0.5 = 89.5 \)
- Upper bound: \( 110 + 0.5 = 110.5 \)

### Step 4: Standardize the values
Now, we calculate the corresponding z-scores for these values using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]

For the lower bound (\(x = 89.5\)):
\[
z_{\text{lower}} = \frac{89.5 - 100}{7.0711} \approx \frac{-10.5}{7.0711} \approx -1.4846
\]

For the upper bound (\(x = 110.5\)):
\[
z_{\text{upper}} = \frac{110.5 - 100}{7.0711} \approx \frac{10.5}{7.0711} \approx 1.4846
\]

### Step 5: Find the probability
Using the standard normal distribution table (or a calculator), we find the probabilities corresponding to these z-scores:
- The probability for \(z_{\text{lower}} = -1.4846\) is approximately 0.0691.
- The probability for \(z_{\text{upper}} = 1.4846\) is approximately 0.9309.

Finally, the probability that between 45% and 55% of the cars roll through the stop sign is:
\[
P(45\% \leq X \leq 55\%) = P(z_{\text{upper}}) - P(z_{\text{lower}}) = 0.9309 - 0.0691 = 0.8618
\]

Thus, the probability is **0.8618** (rounded to four decimal places).

### Conclusion:
The probability that between 45% and 55% of the cars roll through the stop sign is **0.8618**.

Would you like further clarification or additional details? Here are 5 related questions:

1. How does the normal approximation work in statistics?
2. What is a continuity correction and why is it used?
3. Could the sample size affect the accuracy of the normal approximation?
4. How do you interpret z-scores in a normal distribution?
5. What are other real-life applications of the binomial distribution?

**Tip**: In binomial distribution problems, the larger the sample size, the closer the binomial distribution is to the normal distribution!

Discover how to calculate the probability that between 45% and 55% of cars roll through a stop sign using binomial distribution and normal approximation. This detailed solution involves calculating the mean, standard deviation, and applying z-scores, suitable for college-level statistics.

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