Mean Gas Mileage For A Hybrid Car Is 56 Miles Per Gallon

The Mean Gas Mileage For A Hybrid Car Is 56 Miles Per Gallon Suppose

The mean gas mileage for a hybrid car is 56 miles per gallon. Suppose that the gasoline mileage is approximately normally distributed with a standard deviation of 3.2 miles per gallon.

a. What is the probability that a randomly selected hybrid gets more than 60 miles per gallon?

To find this probability, we first calculate the z-score for 60 miles per gallon using the formula:

z = (X - μ) / σ

where X = 60, μ = 56, and σ = 3.2.

z = (60 - 56) / 3.2 = 4 / 3.2 = 1.25

Next, we look up the probability corresponding to z = 1.25 in the standard normal distribution table or use a calculator. The cumulative probability up to z = 1.25 is approximately 0.8944. Since we want the probability of getting more than 60 miles per gallon, we subtract this from 1:

P(X > 60) = 1 - 0.8944 = 0.1056

b. What is the probability that a randomly selected hybrid gets 50 miles per gallon or less?

Calculate the z-score for 50 miles per gallon:

z = (50 - 56) / 3.2 = -6 / 3.2 ≈ -1.875

The cumulative probability for z = -1.875 is approximately 0.0304. Therefore, the probability that a hybrid gets 50 miles per gallon or less is approximately 0.0304.

c. What is the probability that a randomly selected hybrid gets between 57 and 62 miles per gallon?

Calculate z-scores for 57 and 62 miles per gallon:

z1 = (57 - 56) / 3.2 ≈ 0.3125

z2 = (62 - 56) / 3.2 = 6 / 3.2 ≈ 1.875

Lookup cumulative probabilities:

P(z1 ≈ 0.3125) ≈ 0.6230

P(z2 ≈ 1.875) ≈ 0.9699

The probability that the mileage is between 57 and 62 miles per gallon is the difference:

0.9699 - 0.6230 = 0.3469

Therefore, approximately 34.69% of hybrids have gas mileage between 57 and 62 miles per gallon.

d. What is the probability that a randomly selected hybrid gets less than 46 miles per gallon?

Calculate the z-score for 46 miles per gallon:

z = (46 - 56) / 3.2 = -10 / 3.2 ≈ -3.125

The cumulative probability for z = -3.125 is approximately 0.0009.

Thus, the probability that a hybrid gets less than 46 miles per gallon is approximately 0.0009, or 0.09%.

Paper For Above instruction

Understanding the probability distribution of gas mileage in hybrid cars provides valuable insights into their performance variability and reliability. Assuming that the gas mileage follows a normal distribution with a mean of 56 miles per gallon and a standard deviation of 3.2 miles per gallon, we can analyze various probabilities related to this distribution. These probabilities are essential for consumers and manufacturers to assess the likelihood of achieving certain fuel efficiencies and to set realistic expectations.

Firstly, assessing the probability that a randomly selected hybrid exceeds 60 miles per gallon involves calculating the z-score and referencing the standard normal distribution table. The z-score corresponding to 60 miles per gallon is 1.25, leading to a cumulative probability of approximately 0.8944. Subtracting this from 1 yields a probability of about 0.1056, indicating a 10.56% chance that a hybrid exceeds this mileage.

Similarly, evaluating the likelihood of a hybrid obtaining 50 miles per gallon or less involves computing a z-score of approximately -1.875, associated with a cumulative probability of about 0.0304. This suggests a relatively low probability (around 3.04%) that a hybrid’s mileage falls at or below this threshold, reflecting that such low efficiencies are relatively uncommon.

Further, analyzing the probability that a hybrid's mileage falls between 57 and 62 miles per gallon combines two z-scores: 0.3125 and 1.875. The cumulative probabilities associated with these z-scores are roughly 0.6230 and 0.9699, respectively. The difference between these gives a probability of approximately 0.3469, or 34.69%. This quantifies the proportion of hybrids expected to perform within this range, which is valuable for manufacturers in setting performance targets and for consumers in understanding typical vehicle efficiency.

Finally, the probability that a hybrid gets less than 46 miles per gallon is based on a z-score of approximately -3.125, correlating to a cumulative probability of about 0.0009. This indicates an extremely low likelihood (approximately 0.09%) of such a low mileage, underscoring the overall reliability of hybrid vehicles in maintaining a minimum performance level.

These probability assessments exemplify how statistical tools, particularly the normal distribution, are instrumental in evaluating vehicle performance metrics. They enable manufacturers to analyze variability and help consumers make informed decisions based on realistic expectations of hybrid car efficiencies. Moreover, such analyses are fundamental to performance testing, quality assurance, and marketing strategies in the automotive industry, emphasizing the importance of understanding and applying normal distribution concepts in real-world contexts.

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