Question

For a car traveling 30 miles per hour (mph), the distance required to brake to a stop is normally distributed with a mean of 50 feet and a standard deviation of 8 feet. Suppose you are traveling \(30 \mathrm{mph}\) in a residential area and a car moves abruptly into your path at a distance of 60 feet. a. If you apply your brakes, what is the probability that you will brake to a stop within 40 feet or less? Within 50 feet or less? b. If the only way to avoid a collision is to brake to a stop, what is the probability that you will avoid the collision?

Short answer

Answer: The probability of stopping within 40 feet or less is approximately 0.211, within 50 feet or less is approximately 0.500, and within 60 feet or less (avoiding the collision) is approximately 0.789.

Step-by-step solution

Identify the given variables and calculate the z-scores for the given distances

For a normally distributed variable, we have the following information provided: Mean: \(\mu = 50\) feet Standard Deviation: \(\sigma = 8\) feet We need to compute the z-scores for 40 feet and 50 feet. The z-score is calculated using the formula: \(z = \frac{x-\mu}{\sigma}\) where: \(x\) is the distance to be calculated \(\mu\) is the mean \(\sigma\) is the standard deviation Calculating z-scores for 40 feet and 50 feet: \(z_1 = \frac{40 - 50}{8} = -1.25\) \(z_2 = \frac{50 - 50}{8} = 0\)

Find the probability of stopping within 40 feet or less and within 50 feet or less

Next, we need to determine the probability of stopping within 40 feet or less and 50 feet or less. For this, we will use the z-score table to find the area under the curve for each z-score. For \(z_1 = -1.25\), looking up the z-score table, we find the area to the left of the z-score, which is the probability of stopping within 40 feet or less, is approximately 0.211. For \(z_2 = 0\), the area to the left of the z-score is 0.5, which means that there is a 50% probability of stopping within 50 feet or less. #a. The probability of stopping within 40 feet or less is approximately 0.211, and within 50 feet or less is approximately 0.500.

Determine the probability of stopping within 60 feet or less

In order to avoid a collision, we need to determine the probability that the car stops within 60 feet or less. First, let's find the z-score for 60 feet: \(z_3 = \frac{60 - 50}{8} = 1.25\) Now, we need to find the area to the left of the z-score for 60 feet: For \(z_3 = 1.25\), looking up the z-score table, we find the area to the left of the z-score, which is the probability of stopping within 60 feet or less, is approximately 0.789. #b. The probability of stopping within 60 feet or less (avoiding the collision) is approximately 0.789.

Key concepts

Z-Score Calculation

Calculating the z-score is an essential skill when working with normal distributions. It's a way to understand how far away a particular value is from the mean, using the standard deviation as a unit. If you're wondering how to calculate a z-score, it's simple! You use the formula:

• \( z = \frac{x-\mu}{\sigma} \)

Here, \( x \) represents the specific value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
By calculating z-scores, we can convert data to a standardized form and easily compare it to a standard normal distribution.
For example, if we have a distance of 40 feet, mean of 50 feet, and standard deviation of 8 feet, the z-score would be \( \frac{40-50}{8} = -1.25 \). This tells us that 40 feet is 1.25 standard deviations below the mean.
Z-score calculations allow us to find the probability of a value being below or above a certain point in a normal distribution.

Probability

When dealing with normal distributions, calculating probabilities lets us understand how likely certain outcomes are.
After calculating the z-score, the next step is finding the probability using a z-score table, which shows the probability that a value is less than or equal to the z-score. These tables provide the area under the normal curve to the left of a given z-score.

• For a z-score of -1.25, the probability is approximately 0.211, meaning there's a 21.1% chance of stopping within 40 feet.
• For a z-score of 0, the probability is 0.5, indicating a 50% chance of stopping within 50 feet.
• For a z-score of 1.25, the probability is approximately 0.789, showing a 78.9% chance of stopping within 60 feet.

Understanding these probabilities can help make informed decisions, such as knowing the likelihood of stopping safely and avoiding a collision.

Standard Deviation

The standard deviation is a key concept in understanding the spread or dispersion of a set of values in a dataset. It measures how much the values in a distribution deviate from the mean. A small standard deviation means values are clustered close to the mean, while a larger standard deviation indicates more spread.

• In our exercise, the standard deviation is 8 feet. This gives insight into how much the braking distances vary from the average distance of 50 feet.

Calculating standard deviation involves:

• Finding the mean of all values.
• Subtracting the mean from each value to find the deviation of each.
• Squaring these deviations to make them positive.
• Averaging the squared deviations.
• Taking the square root of this average.

By understanding standard deviation, we gain insight into the reliability of our calculated probabilities and predictions within the context of a normal distribution.