The tread life of Stone Soup tires can be modeled by a...

Name: Problem 48, Chapter 5: Statistics for Business and Economics
Uploaded: 2022-01-19T19:50:46-08:00
Duration: 1 s
Channel: Sheryl Ezze
Description: The tread life of Stone Soup tires can be modeled by a normal distribution with a mean of 35,000 miles and a standard deviation of 4,000 miles. A sample of 100 of these tires is taken. What is the probability that more than 25 of them have tread lives of more than 38,000 miles?

The tread life of Stone Soup tires can be modeled by a normal distribution with a mean of 35,000 miles and a standard deviation of 4,000 miles. A sample of 100 of these tires is taken. What is the probability that more than 25 of them have tread lives of more than 38,000 miles?

Key Concepts

Standardization (Z-scores)

Standardization involves converting a normal random variable to a standard normal variable (Z-score) by subtracting the mean and dividing by the standard deviation. This process enables the use of standard normal distribution tables to determine probabilities associated with tire tread lives exceeding a specified threshold.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, mean, and standard deviation. In this context, it is used to model the tire tread life with a specified mean and variability, allowing for the calculation of probabilities for tire lifetimes exceeding a certain value.

Binomial Distribution

The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success. Here, each tire represents a trial where the success condition is a tread life exceeding 38,000 miles, allowing the count of such tires in a sample to be modeled by a binomial distribution.

Normal Approximation to the Binomial

For large sample sizes, the binomial distribution can be approximated by a normal distribution when the success probability is not too close to 0 or 1. This approximation simplifies the calculation of probabilities, such as finding the likelihood that more than a specified number of tires have tread lives exceeding the given threshold.

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