Based on tests of the Chevrolet Cobalt, engineers have found that the miles per gallon in highway driving are normally distributed, with a mean of 33 miles per gallon and a standard deviation of 4 miles per gallon. Round your answers to 4 decimal places. (a) What is the probability that a randomly selected Cobalt gets more than 34 miles per gallon? (b) If sixteen Cobalts are randomly selected and the miles per gallon for each car are recorded, what is the probability that the mean miles per gallon exceeds 34 mpg? (c) Could you have computed the probability in part (b) if you were not told that the miles per gallon of a randomly selected Cobalt are normally distributed? No, we could not, since we couldn't use Central Limit Theorem because the sample size is only 16. Yes, we could, because the sample size is large enough and we could use the Central Limit Theorem. Yes, we could, because the distribution of the sample mean is always normal regardless of the sample size by CLT.

Question

Kara Daniel

Mathematics

24 December, 20:33

Based on tests of the Chevrolet Cobalt, engineers have found that the miles per gallon in highway driving are normally distributed, with a mean of 33 miles per gallon and a standard deviation of 4 miles per gallon. Round your answers to 4 decimal places. (a) What is the probability that a randomly selected Cobalt gets more than 34 miles per gallon? (b) If sixteen Cobalts are randomly selected and the miles per gallon for each car are recorded, what is the probability that the mean miles per gallon exceeds 34 mpg? (c) Could you have computed the probability in part (b) if you were not told that the miles per gallon of a randomly selected Cobalt are normally distributed? No, we could not, since we couldn't use Central Limit Theorem because the sample size is only 16. Yes, we could, because the sample size is large enough and we could use the Central Limit Theorem. Yes, we could, because the distribution of the sample mean is always normal regardless of the sample size by CLT.

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Answers (1)

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Gilberto Bartlett · Answer 1

Step-by-step explanation:

Let x be the random variable representing the miles per gallon of Chevrolet Cobalt in highway driving. Since it is normally distributed and the population mean and population standard deviation are known, we would apply the formula,

z = (x - µ) / σ

Where

x = sample mean

µ = population mean

σ = standard deviation

From the information given,

µ = 33

σ = 4

a) the probability that a randomly selected Cobalt gets more than 34 miles per gallon is expressed as

P (x > 34) = 1 - P (x ≤ 34)

For x = 34,

z = (34 - 33) / 4 = 0.25

Looking at the normal distribution table, the probability corresponding to the z score is 0.5987

P (x > 34) = 1 - 0.5987

P (x > 34) = 0.4013

b) since sample size is given, the formula would be

z = (x - µ) / (σ/n)

n = 16

z = (34 - 33) / (4/√16) = 1

Looking at the normal distribution table, the probability corresponding to the z score is 0.8413

P (x > 34) = 1 - 0.8413

P (x > 34) = 0.1587

c) Since the population is normally distributed, then, the sample is also normally distributed. The correct option is

Yes, we could, because the distribution of the sample mean is always normal regardless of the sample size