7. The average price of a gallon of unleaded regular gasoline was reported to be

Question

7. The average price of a gallon of unleaded regular gasoline was reported to be $3.13 in Illinois (http://www.cincygasprices.com). Use this price as the population mean, and assume the population standard deviation is $.20. [45 pts]

a. Assume a sample of 30 service stations is collected, what is the value of the standard error of the mean? [5 pts]

b. What is the probability that the mean price for a sample of 30 service stations is within

$.03 of the population mean? [10 pts]

c. What is the probability that the mean price for a sample of 50 service stations is within

$.03 of the population mean? [10 pts]

d. What is the probability that the mean price for a sample of 100 service stations is within

$.03 of the population mean? [10 pts]

e.         What is the probability that the mean price for a sample of 100 service stations is between $3 and $3.2?

Explanation / Answer

a) Standard error of mean = 0.20/Sqrt(30) = 0.0365

b) P(3.13-0.03<X<3.13+0.03)

= P((-0.03/0.20)*Sqrt(30)<Z<(0.03/0.20)*Sqrt(30))

= P(-0.82158<z<0.82158)

= 0.5887(rounded to four decimal places)

c) P(3.13-0.03<X<3.13+0.03)

= P((-0.03/0.20)*Sqrt(50)<Z<(0.03/0.20)*Sqrt(50))

= P(-1.06066<z<1.06066)

= 0.7112 (rounded to four decimal places)

d) P(3.13-0.03<X<3.13+0.03)

= P((-0.03/0.20)*Sqrt(100)<Z<(0.03/0.20)*Sqrt(100))

= P(-1.5<z<1.5)

= 0.8664 (rounded to four decimal places)

e) P(3<X<3.2)

= P((3-3.13/0.20)*Sqrt(100)<z<(3.2-3.13/0.20)*Sqrt(100))

= P(-6.5<z<3.5)

= 0.9998 (rounded to four decimal places)

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