According to an automobile association, the average cost of a gallon of regular

Question

According to an automobile association, the average cost of a gallon of regular unleaded fuel at gas stations in a certain month was $2.826. Assume that the standard deviation of such costs is $0. 15 Suppose that a random sample of n = 100 gas stations is selected from the population and the month's cost per gallon of regular unleaded fuel is determined for each Consider x. the sample mean cost per gallon. Complete through d. Click the icon to view the table of normal curve areas Calculate mu_x^- and sigma_x^- Do not include the $ symbol in your answer Round to three decimal places as needed) (Do not include the S symbol in your answer Round to three decimal places as needed) What is the approximate probability that the sample has a mean fuel cost between $2.84 and $2.86?

Explanation / Answer

Given:

Mean price µ = $2.826

Standard deviation = $0.15

A random sample of 100 stations is considered

Hence sample size n = 100

Since it is not mentioned that the population follows Normal distribution, Central limit theorem can be applied which states that if a sample is drawn from any unknown population it will follow normal distribution with mean xbar equal to population mean µ & sample standard deviation s equal to population standard deviation divided by root of sample size n, provided the sample drawn isa large sample

Hence as n> 30, we can apply central limit theorem and get the answers for parta:

Part a)

µxbar = µ [ as per central limit theorem, sample mean = population mean]

Hence µxbar = $2.826

xbar = /n [as per central limit theorem sample standard deviation s = population standard deviation/sample size

Hence xbar = 0.15/100 = 0.015

xbar = 0.015

Part b)

P(2.84 xbar 2.86) = P(xbar 2.86) - P(xbar 2.84) [equation I]

To find P(xbar 2.86) , we need to find the corresponding Z score (z1)

z1 = (x - xbar)/s

s = 0.015

xbar = 2.826

x = 2.86

z1 = (2.86-2.826)/0.015 = 2.2667

Now we need to refer the Z table to get the value corresponding to z1 = 2.2667

Hence P(z1 2.2667) = 0.988296 [equation II]

Similarly we need to find Z score (z2) in order to find P(x 2.84)

z2 = (x - xbar)/s

s = 0.015

xbar = 2.826

x = 2.84

z2 = (2.84-2.826)/0.015 = 0.9333

Now we need to refer the Z table to get the value corresponding to z2 = 0.9333

Hence P(z2 0.9333) = 0.824667 [equation III]

Finally plug in equation II & III, in equation I, we get :

P(2.84 xbar 2.86) = 0.988296 - 0.824667

P(2.84 xbar 2.86) = 0.163628

Since we need to round off the answer to four decimal places

P(2.84 xbar 2.86) = 0.1636

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.