Question

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 100 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.)

Answer 1

To calculate the probability that between 45% and 55% of the cars roll through the stop sign, we can use the normal approximation to the binomial distribution. The probability is approximately 0.6827.

Step-by-step explanation:

To calculate the probability that between 45% and 55% of the cars roll through the stop sign, we need to use the normal approximation to the binomial distribution. Given that half of all cars roll through the stop sign without stopping, the probability of a car rolling through the stop sign is 0.5. We can calculate the mean and standard deviation of the distribution using the formulas:

Mean: np = 100 * 0.5 = 50

Standard Deviation: sqrt(np(1-p)) = sqrt(100 * 0.5 * (1-0.5)) = 5

Next, we can convert the range of 45% to 55% into z-scores using the formula:

Z-Score: (x - mean) / standard deviation

For 45%, the z-score is (45 - 50) / 5 = -1, and for 55%, the z-score is (55 - 50) / 5 = 1.

Using a standard normal distribution table or a calculator, we can find the cumulative probability associated with each z-score. Subtraction the probability associated with the lower z-score from the probability associated with the higher z-score gives us the probability that between 45% and 55% of the cars roll through the stop sign.

The probability is approximately 0.6827.