Forensic researchers at a vehicle safety research center were interested in empi
ID: 3222426 • Letter: F
Question
Forensic researchers at a vehicle safety research center were interested in empirically studying the stopping distance required for vehicles. Two main contributors to stopping distance are: (i) speed of the vehicle and (ii) road condition (e.g., dry asphalt, wet asphalt, dry concrete, wet concrete, snow, ice). The contributors were controlled at a speed of 75 mph and a dry asphalt road condition. Assume the "imperial" units of measure, which require speed in mph and stopping distance in feet. The following are observations of stopping distance under the above condition for seven vehicles. 261 275 242 251 254 249 268 a. What does "empirical" mean in the context of experimentation? b. Is there sufficient evidence to conclude that vehicles exceed a mean stopping distance of 250 feet? Use the critical value approach with significance of 0.1. c. Find the value of beta when the true mean is 257.5 feet. d. Find the value of beta when the true mean is 280 feet.Explanation / Answer
a. Empiricla mean here in the context of experimentation = mean of al given seven values = 257.1429 feet
standard deviation of the sample s = 11.51 feet
b. Null hypothesis : Ho : stopdistance = 250 feet.
Alternative Hypothesis : H1: stopdistance > 250 feet.
significane level = 0.1 and it is right tailed test
Test Statisitics
t = ( xbar- 250)/ (s/ n) = ( 257.1429 - 250)/ ( 11.51/ 7) = 7.1429/4.35 = 1.642
for dF = 7-1 = 6 and for significance value alpha = 0.1 => tcritical = 1.440
so t > tcritical, that means that there is sufficient evidence to conclude that vehicles exceed a mean stopping distance of 250 feet.
(c) value of when true mean is 257.5 feet.
Now we will calculate the upper bound of mean stopping distance which is equal to
q = 250 + 1.440 * (11.51/7) = 256.265
The true mean 0= 257.5 feet
so here we have to calculate Probability that we will not reject the null hypothesis even when it is false.
so we will calculate Pr( xbar<= 256.265 ; 257.5; 11.51/7 )
so Z - value = (256.265 - 257.5)/ (11.51/7 ) = ( - 1.235)/ 4.35 = -0.2830
so P -value = 0.3885
so = 0.3885
(d) value of when true mean is 280 feet.
Now we will calculate the upper bound of mean stopping distance which is equal to
q = 250 + 1.440 * (11.51/7) = 256.265
The true mean 0= 280 feet
so here we have to calculate Probability that we will not reject the null hypothesis even when it is false.
so we will calculate Pr( xbar<= 256.265 ; 280; 11.51/7 )
so Z - value = (256.265 - 280)/ (11.51/7 ) = ( - 23.735)/ 4.35 = -10.91
so P -value = 0
so = 0