Vehicle speeds at a certain highway location are believed to follow a normal dis

Question

Vehicle speeds at a certain highway location are believed to follow a normal distribution with mean µ = 60 mph and standard deviation = 6 mph. The speeds for a randomly selected sample of n = 23 vehicles will be recorded.

(a) Give numerical values for the mean and standard deviation of the sampling distribution of possible sample means for randomly selected samples of n = 23 from the population of vehicle speeds.

(b) Does the sampling distribution of the possible sample means have an approximate normal distribution? Explain.

(c) Use the Empirical Rule to find values that fill in the blanks in the following sentence: For a random sample of n = 23 vehicles, there is about a 95% chance that the mean vehicle speed in the sample will be between and mph.

(d) Sample speeds for a random sample of 23 vehicles are measured at this location, and the sample mean is 66 mph. Given the answer to part (c), explain whether this result is consistent with the belief that the mean speed at this location is µ = 60 mph.

Explanation / Answer

a)

By central limit theorem, the mean is the same,

u(X) = 60 mph [ANSWER]

and the standard error is reduced,

sigma(X) = sigma/sqrt(n) = 6/sqrt(23) = 1.251086484 mph [ANSWER]

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b)

YES, because the original distribution is already normal.

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c)

It is within 2 standard errors from the mean, so between

u - 2*sigma = 60 - 2*1.251086484 = 57.49782703
u + 2*sigma = 60 + 2*1.251086484 = 62.50217297

Hence, it is between these two numbers.

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d)

We first get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    66      
u = mean =    60      
n = sample size =    23      
s = standard deviation =    6      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    4.795831523      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   4.795831523   ) =    8.10007E-07

As this is a very low probability, then NO, THIS IS NOT CONSISTENT.

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