How to answer in a correct methodology

Question

The speed of vehicles passing through the construction zone are normally distributed with a population mean of 46 miles per hour and a standard deviation of 4 miles per hour. What is the probability that a randomly selected vehicles passing through this construction zone are?

a) Between 50 and 55 miles per hour?                                                                   (1 Mark)

b) Exactly 46 miles per hour?                                                                                  (1 Mark)

c)  Exceeding 49 miles per hour?                                                                            (1 Mark)

d) Less than 50 miles per hour?                                                                              (1 Mark)

e) Between 40 and 50 miles per hour?                                                                   (1 Mark)

Answer 1

Although this question can be answered in a couple of minutes by using the normcdf function on a graphing calculator (such as a TI-84), you seem to want to know how to solve it using basic concepts. To do this requires finding the z-score of each specific speed mentioned, using the formula z(x) = (x - μ)/σ, where x is the specific speed, μ is the average speed and σ is the standard deviation, Applying this formula yields:

z(40) = (40 - 46)/4 = -1.5

z(46) = (46 - 46)/4 = 0

z(49) = (49 - 46)/4 = 0.75

z(50) = (50 - 46)/4 = 1

z(55) = (55 - 46)/4 = 2.25

The next step is to use the table of the normal distribution to find the probability, for each z-value, of that value or any lower value occurring. Let's express that in the form P(-∞, z).

P(-∞, -1.5) = .0668

P(-∞, 0) = .5000

P(-∞, 0.75) = .7734

P(-∞, 1) = .8413

P(-∞, 2.25) = .9878

We could also add in P(-∞, ∞) = 1.0000

a. I'll continued to use the z-scores as the second variable in the probability function, but it could be done using the speed values that correspond to each z-score.

P(1, 2.25) = P(-∞, 2.25) - P(-∞, 1) = .9878 - .8413 = .1465

b. For a continuous distribution, because there are an infinite number of possibles values, the probability of any exact value is zero.

We could also say P(0, 0) = P(-∞, 0) - P(-∞, 0) = .5000 - .5000 = 0

c. P(.75, ∞) = P(-∞, ∞) - P(-∞, .75) = 1.0000 - .7734 = .2266

d. P(-∞, 1) = .8413

e. P(-1.5, 1) = P(-∞, 1) - P(-∞, -1.5) = .8413 - ..0668 = .7745

Answer 2

Let x be the speed, x is normally distributed with mean 46 and sd 4.

z = (x - 46) / 4 is normally distributed with mean 0 and sd 1 and can be used to look up probability in standard normal probability table.

a) We want to find P (50 < x < 55). We convert that to z:

P( (50 - 46)/4 < z < (55-46)/4) = P(0.25 < z < 2.25) = P(z < 2.25) - P(z < 0.25)

Look up those probabilities using standard normal table of calculator function

b) since normal distribution is continuous the probability it takes on a single value is 0. Can only determine probability for a range of values.

c) P(x > 49) = P(z > (49-46)/4) = P(z > 0.75) = 1 - P(z < 0.75)

d) P(x < 50) = P(z < (50-46/4) = P (z < 1)

e) We want to find P (40 < x < 50).

P( (40 - 46)/4 < z < (50-46)/4) = P(-1.5 < z < 1) = P(z < 1) - P(z < -1.5)