Problem 9 - Applied Calculus

Question

The speeds of cars on a road are approximately normally distributed with a mean \(\mu=58 \mathrm{km} / \mathrm{hr}\) and standard deviation \(\sigma=4 \mathrm{km} / \mathrm{hr}\) (a) What is the probability that a randomly selected car is going between 60 and \(65 \mathrm{km} / \mathrm{hr} ?\) (b) What fraction of all cars are going slower than 52 \(\mathrm{km} / \mathrm{hr} ?\)

Step-by-Step Solution

Answer

(a) 26.84% of cars are between 60 and 65 km/hr. (b) 6.68% of cars are slower than 52 km/hr.

Step 1: Standardize the Distribution

To find probabilities for normally distributed data, we convert the problem to the standard normal distribution. The formula to find the standard score (z-score) is: \[ z = \frac{X - \mu}{\sigma} \] where \(X\) is the value in the distribution, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

Step 2: Calculate Z-scores for 60 and 65 km/hr

First, calculate the z-score for 60 km/hr: \[ z_{60} = \frac{60 - 58}{4} = \frac{2}{4} = 0.5 \] Now, calculate the z-score for 65 km/hr: \[ z_{65} = \frac{65 - 58}{4} = \frac{7}{4} = 1.75 \]

Step 3: Find Probabilities Using Z-table

We need to find the probability that a car's speed is between 60 and 65 km/hr. Use the z-table to find the probabilities for z-scores.- Probability that z is less than 0.5: approximately 0.6915- Probability that z is less than 1.75: approximately 0.9599The probability that a car's speed is between 60 and 65 km/hr is:\[ P(60 < X < 65) = P(Z < 1.75) - P(Z < 0.5) = 0.9599 - 0.6915 = 0.2684 \]

Step 4: Calculate Z-score for 52 km/hr

Calculate the z-score for 52 km/hr: \[ z_{52} = \frac{52 - 58}{4} = \frac{-6}{4} = -1.5 \]

Step 5: Find Probability for Speed Slower than 52 km/hr

Using the z-table for z = -1.5, the probability that a car is slower than 52 km/hr is approximately 0.0668.

Key Concepts

Understanding Z-score Calculation

Z-score calculation is a crucial step when working with a normal distribution. It allows you to standardize your data, which means you can compare values from different normal distributions. A z-score tells you how many standard deviations a value, or score, is from the mean. The formula used is:

• \( z = \frac{X - \mu}{\sigma} \)

where \(X\) is the data value, \(\mu\) is the mean of the dataset, and \(\sigma\) represents the standard deviation.

By calculating the z-score, you translate the problem that involves specific data values into a problem that involves the standard normal distribution. When you plug in the numbers from our exercise, you'll find that a speed of 60 km/hr translates to a z-score of 0.5 and 65 km/hr translates to 1.75. These scores show how far these speeds are from the average speed of 58 km/hr.

Exploring Probability in Normal Distribution

Probability in the context of normal distribution is about figuring out the likelihood that a certain event will occur. Specifically, when we talk about the speeds of cars fitting a normal distribution, probability helps us determine how many cars, on average, will be within a certain speed range. To solve this, we often use the z-score values to look up standard normal distribution tables, also known as z-tables.

A z-table shows the probability of a z-score being below a certain value. So, for our exercise, you look up the values for the z-scores of 60 and 65 km/hr.

• The probability for 60 km/hr (z=0.5) was found to be about 0.6915.
• For 65 km/hr (z=1.75), it was about 0.9599.

By subtracting these probabilities, you find the probability of a car traveling between these two speeds. This subtraction process gives you a clearer picture of the likelihood of cars falling within a specific speed range.

Decoding Standard Deviation

Standard deviation is a essential concept in statistics that quantifies the amount of variation or dispersion in a set of data values. In simpler terms, it tells you how spread out the numbers in a dataset are around the mean.

A smaller standard deviation implies that the numbers are close to the mean, indicating consistency. On the other hand, a larger standard deviation suggests more variance, meaning the data is more spread out.

• In our exercise, the standard deviation of the car speeds is 4 km/hr.

With this standard deviation, you can calculate how the speeds of the cars deviate from the average speed of 58 km/hr. It is this metric that forms the "denominator" in the z-score calculation, standardizing diverse datasets onto a single common scale.

Mean Calculation: The Center of Data

The mean, often referred to as the average, is the total sum of all values in a data set divided by the number of values. It serves as a measure of the center in a data distribution. Calculating the mean gives a representation of the dataset, acting as a balance point.

In a normal distribution scenario, such as the speed of cars in our example, the mean provides a point from which we can determine deviation or differences using z-scores.

• For our exercise, the mean speed is 58 km/hr.

Knowing this mean is fundamental, as it allows us to understand and calculate probabilities for any given car speed when combined with the standard deviation. It's the starting point in figuring out how likely it is for cars to be traveling at different speeds on the road.