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Chapter 8: Problem 20

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 \(\mathrm{km} / \mathrm{hr} ?\)

Short Answer

Expert verified
(a) 0.2684; (b) 0.0668

Step by step solution

01

Identify the problem and given values

This problem requires us to find probabilities for a normally distributed dataset. We are provided with a mean \(\mu = 58\, \text{km/hr}\) and a standard deviation \(\sigma = 4\, \text{km/hr}\).
02

Calculate Z-scores for 60 and 65 km/hr

The \( Z \)-score for a value \( X \) in a normal distribution is calculated as:\[ Z = \frac{X - \mu}{\sigma} \]First, calculate the \( Z \)-score for 60 km/hr:\[ Z_{60} = \frac{60 - 58}{4} = 0.5 \]Next, calculate the \( Z \)-score for 65 km/hr:\[ Z_{65} = \frac{65 - 58}{4} = 1.75 \]
03

Find probabilities using Z-table for part (a)

Using a standard normal distribution table, find the probabilities for \( Z_{60} = 0.5 \) and \( Z_{65} = 1.75 \):- \( P(Z < 0.5) \approx 0.6915 \)- \( P(Z < 1.75) \approx 0.9599 \)The probability that a car's speed is between 60 and 65 km/hr is:\[ P(60 < X < 65) = P(Z < 1.75) - P(Z < 0.5) = 0.9599 - 0.6915 = 0.2684 \]
04

Calculate Z-score for 52 km/hr

The \( Z \)-score for 52 km/hr is:\[ Z_{52} = \frac{52 - 58}{4} = -1.5 \]
05

Find probability for Z-score of 52 km/hr

Using a standard normal distribution table, find the probability for \( Z_{52} = -1.5 \):- \( P(Z < -1.5) \approx 0.0668 \)Thus, the fraction of cars going slower than 52 km/hr is approximately 6.68%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Calculation
Understanding probability in the context of a normal distribution helps us find the likelihood of an event occurring within a specific range. In a normal distribution, probabilities are calculated using the area under the curve. To find these probabilities, we often rely on the standardized normal distribution, which is also known as the z-distribution. This means converting the values in question to z-scores, allowing us to use standard normal distribution tables to find probabilities.
  • This involves determining the probability of an event by finding the area under the curve between two points.
  • Using tables or statistical software helps us find how much area or probability lies between specified data points.
When calculating probabilities, it's essential to convert our data points into z-scores. This ensures that different normal distributions are comparable.
Z-Score
A z-score is a measure that expresses the number of standard deviations a data point is from the mean. Calculating the z-score allows us to understand how unusual or typical a certain result is. The formula for the z-score is:\[ Z = \frac{X - \mu}{\sigma} \]where \(X\) is the value for which we're calculating the z-score, \(\mu\) is the mean, and \(\sigma\) is the standard deviation of the dataset.
  • The z-score helps standardize different datasets, enabling us to compare values across different normal distributions.
  • A positive z-score denotes a value above the mean, while a negative z-score indicates a value below the mean.
  • Once the z-score is determined, we can use the standard normal distribution table to find the associated probability.
Z-scores play a crucial role in statistical analysis by allowing interpretable and meaningful probability calculations.
Standard Deviation
Standard deviation is a statistic that measures the amount of variation or dispersion in a set of data. In the context of a normal distribution, it tells us how spread out the values in the dataset are around the mean. A small standard deviation signifies that the data points tend to be close to the mean, while a large standard deviation indicates that the data points are spread out over a broader range of values.
  • It's calculated as the square root of the average of the squared deviations from the mean.
  • In practice, the standard deviation allows us to determine whether a given value is far from the mean or within a typical range.
  • In a normal distribution, about 68% of values lie within one standard deviation of the mean.
Understanding standard deviation is essential for assessing data normality and making sound predictions based on the data.
Mean
The mean, often called the average, is a measure of central tendency that is crucial for statistical analysis. It represents the sum of all values in a dataset divided by the number of values. In a normal distribution, the mean is at the center of the symmetry.
  • It is calculated by adding all values together and then dividing by the total number of values.
  • As the central value, the mean is used to compute the z-scores, which help standardize and evaluate data points.
  • It is important to note that the mean may be affected by extreme values or outliers.
Having a clear understanding of the mean is fundamental to interpreting data and using statistical measures effectively.

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