I need help with this homework. Please can you also show me how to get the answe
ID: 3355659 • Letter: I
Question
I need help with this homework. Please can you also show me how to get the answer using excel spreadsheet FORMULA?
0.8729
QUESTIONS 3-9 ARE BASED ON THE FOLLOWING INFORMATION. Assume the vehicle speed on a freeway has a bell-shaped distribution with mean speed of 82 mph and standard deviation of 7.4 mph. 3 A single vehicle is randomly clocked. The probability that the vehicle speed is below the speed limit of 80 mph is: a 0.4338 b 0.4131 c 0.3935 0.3935 d 0.3738 4 The probability that the vehicle speed is within ±3 mph from the mean is: a 0.3809 b 0.3623 0.34259 0.65741 c 0.3422 0.657410231 -0.34259 d 0.3148 0.31482 5 The middle interval which includes the speed of 95 percent of all vehicles on the freeway is, a 64.8 99.2 b 67.5 96.5 c 69.8 94.2 67.50 d 71.3 92.7 96.50 6 Instead of a single vehicle, 16 vehicles are randomly clocked. The probability that the mean speed of the 16 vehicles is below 80 mph is. a 0.1398 b 0.1258 c 0.1133 d 0.1019 7 If you repeatedly clock 16 vehicles at a time and obtain the mean of each sample, the fraction of sample means that would fall within ±3 mph from the mean speed of all vehicles is ______. a 0.7919 b 0.8249 c 0.8593 d 0.8951 8 With the same sample size as in the previous problem, the middle interval which contains 95% of the mean vehicle speed of all such samples is _____. a 81.1 82.9 b 80.2 83.8 c 79.3 84.7 78.37 d 78.4 85.6 85.63 9 Assume the population mean price per gallon of gasoline is = $2.35, and the population standard deviation is = $0.10. Suppose that a random sample of 30 gasoline stations will be selected. What is the probability that the simple random sample will provide a sample mean within 3¢ ($0.03) of the population mean? a 0.9316 b 0.9108 c 0.8997 d0.8729
Explanation / Answer
Solution
Let X = vehicle speed on a freeway.
We ~ N(µ, 2), where µ = 82 mph and = 7.4 mph
Back-up Theory
If a random variable X ~ N(µ, 2), i.e., X has Normal Distribution with mean µ and variance 2, then, pdf of X, f(x) = {1/(2)}e^-[(1/2){(x - µ)/}2] …………………………….(A)
Z = (X - µ)/ ~ N(0, 1), i.e., Standard Normal Distribution ………………………..(1)
P(X or t) = P[{(X - µ)/ } or {(t - µ)/ }] = P[Z or {(t - µ)/ }] .………(2)
X bar ~ N(µ, 2/n),…………………………………………………………….…….(3),
where X bar is average of a sample of size n from population of X.
So, P(X bar or t) = P[Z or {(n)(t - µ)/ }] …………………………………(4)
Probability values for the Standard Normal Variable, Z, can be directly read off from
Standard Normal Tables or can be found using Excel Function……………………..(5)
Q3
Probability that the speed of a randomly clocked vehicle is below 80 mph
= P(X < 80)
= P[Z < {(80 - 82)/7.4}] [vide (1) of Back-up Theory]
= P(Z < - 0.2703)
= 0.3935 [using Excel Function on Normal Distribution] ANSWER
Q4
Probability that the vehicle speed is within ± 3 mph from the mean
= P(79 < X < 85)
= P[{(79 - 82)/7.4} < Z < {(85 - 82)/7.4}] [vide (1) of Back-up Theory]
= P(- 0.405 < Z < 0.405)
= P(|Z| < 0.405)
= 0.3146 [using Excel Function on Normal Distribution] ANSWER
Q5
If (t1, t2) be the middle interval holding the speed of 95% of all vehicles, then by symmetry property of Normal Distribution, t1 = - t2. Thus, we have
P(|X| < t) = 0.95 or
P[|Z| < {(t- 82)/7.4}] = 0.95
=> {(t- 82)/7.4} = 1.96 [using Excel Function on Normal Distribution]
Or t = 82 ± 14.504
= (67.496, 96.504)
Thus the middle interval which holds the speed of 95% of all vehicles is
(67.496 mph, 96.504 mph) ANSWER
Q6
Probability mean speed of 16 randomly picked vehicles is below 80 mph
= P(Xbar < 80)
= P[Z < {(80 - 82)/(7.4/4)}] [vide (1), (3) and (4) of Back-up Theory]
= P(Z < - 1.081)
= 0.1398 [using Excel Function on Normal Distribution] ANSWER
DONE