Question 1199323: major automobile company claims that its new model has an average rating of 25 mpg (miles per gallon). Company officials concede that some cars vary based on a variety of factors, and that the mpg performances have a standard deviation of 4 mpg. You are employed by a consumer protection group that routinely test-drives cars. Taking five cars at random off the assembly line, your group finds them to have a poor mpg performance defined as 20 mpg or below.
a. Assuming the company’s claim to be true (μ = 25 and σ = 4), what is the probability that a single car selected randomly performs poorly (mpg of 20 or below)?
b. Assuming the company’s claim to be true (μ = 25 and σ = 4), what is the probability that five cars selected randomly all perform poorly (mpg of 20 or below)?
c.iven the poor performance that your group observed with the five test cars, what conclusion can you draw about the company’s mpg claim?
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! **a. Probability of a Single Car Performing Poorly**
* **Standardize the value:**
* z = (X - μ) / σ
* z = (20 - 25) / 4 = -1.25
* **Find the probability using a standard normal distribution table or calculator:**
* P(X ≤ 20) = P(Z ≤ -1.25) ≈ 0.1056
* **Therefore, the probability that a single randomly selected car has an mpg of 20 or below is approximately 0.1056 (or 10.56%).**
**b. Probability of All Five Cars Performing Poorly**
* **Assuming independent events:** Since the performance of each car is independent of the others, the probability of all five cars having an mpg of 20 or below is:
* P(all five cars ≤ 20 mpg) = P(car 1 ≤ 20 mpg) * P(car 2 ≤ 20 mpg) * ... * P(car 5 ≤ 20 mpg)
* P(all five cars ≤ 20 mpg) = (0.1056)⁵ ≈ 0.000012
* **Therefore, the probability that all five randomly selected cars have an mpg of 20 or below is approximately 0.000012 (or 0.0012%).**
**c. Conclusion**
The probability of observing five cars with an mpg of 20 or below, assuming the company's claim is true, is extremely low (0.0012%). This suggests that:
* **The company's claim of an average mpg of 25 might be inaccurate.**
* **There might be a problem with the production process or the specific batch of cars tested.**
**Further Investigation:**
* **Larger Sample Size:** A larger sample size would provide more robust evidence to support or refute the company's claim.
* **Statistical Testing:** A more formal statistical test, such as a hypothesis test, could be conducted to determine the statistical significance of the observed results.
**Disclaimer:** This analysis assumes that the mpg of the cars follows a normal distribution.
**Note:** This analysis provides a basic framework. A more rigorous analysis would require considering factors like sampling bias, potential measurement errors, and the specific conditions under which the mpg tests were conducted.
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