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Question

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 3.3 miles per gallon. (a) What proportion of hybrids gets over 62 miles per gallon? (b) What proportion of hybrids gets 50 miles per gallon or less? (c) What proportion of hybrids gets between 57 and 62 miles per gallon? (d) What is the probability that a randomly selected hybrid gets less than 46 miles per gallon?

Answer 1

You have to bring this question into the z-score context where you have to use the formula z = (X-μ)/σ. In this question, μ = 56 and σ = 3.3. Particular miles per gallon in smaller questions (a) - (d) should be Xs. Once you plug in X, μ, and σ, and then, simplify the numbers in the formula, you will get z-scores for each. For example, MPG of 62 in question (a) would be z = (62-56)/3.3 = 1.82 (rounded up). Because the question statement (a) says, "proportion over 62 MPG," you are now trying to get the proportion of z-scores greater than 1.82. From the unit normal table that you may have, you can get the proportion by yourself.

Question (b) would be the same proportion with the answer for (a), because the z-score for MPG of 50 has the same magnitude but a negative z-score (-1.82). The proportion that is being asked is "50 MPG or less," that is the same proportion for (a), because the distribution of MPG values and z-scores are considered approximately normal.

Question (c) asks you to get a part of proportion in the right side of distribution (57 and 62 MPG values are all bigger than 56 MPG as the mean). So, you can get the target proportion by subtracting the proportion over 62 MPG (you already got it from (a)) and the proportion less than 57 MPG from 1.00. In other words, an arithmetic operation of "1 - (proportion over z = 1.82) - (proportion less than z = 0.30)" will give you the target proportion.

Question (d) is the same type of question about a specific proportion involved in proportion in (a) or (b), because the probability and proportion is the same concept in this context.

Hope this help you better understand the context and practice more.