Suppose that we would like to determine if the average traffic flow at an inters

Question

Suppose that we would like to determine if the average traffic flow at an intersection in the city is greater than 50 cars per minute. A set of 49 one-minute observation intervals over a one-week period was randomly selected and the average traffic flow was observed to be 47.60 cars/minute. Assume that the standard deviation is known to be 7 cars/minute.

(a) Using a 0.02 level of significance, would you reject the hypothesis that the average traffic flow is greater than 50 cars/minute? What is the Type I error?

(b) If the true average traffic flow is 48 cars/minute, with the test in (a), what is the Type II error? (The probability that we will not reject the hypothesis that the average traffic flow is greater than 50 cars/minute).

Explanation / Answer

a)

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   50  
Ha:    u   <   50  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical z, as alpha =    0.02   ,      
alpha =    0.02          
zcrit =    -   2.053748911      
              
Getting the test statistic, as              
              
X = sample mean =    47.6          
uo = hypothesized mean =    50          
n = sample size =    49          
s = standard deviation =    7          
              
Thus, z = (X - uo) * sqrt(n) / s =    -2.4          
              
Also, the p value is              
              
p =    0.008197536          
              
As |z| > 2.053, and P < 0.02, we   REJECT THE NULL HYPOTHESIS.      

There is significant evidence that the true average traffic flow is less than 50 cars per minute. [CONCLUSION]

A type I error here is incorrectly concluding that the true average traffic flow is less than 50 cars per minute, when in fact, it is not. [ANSWER]

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b)

Note that              

Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.02          
X = sample mean =    50          
z(alpha/2) = critical z for the confidence interval =    2.053748911          
s = sample standard deviation =    7          
n = sample size =    49          
              
Thus,              

Lower bound =    47.94625109          

We THEN get the z score for the critical value. As z = (x - u) sqrt(n) / s, then as          
          
x = critical value =    47.94625109      
u = mean =    48      
n = sample size =    49      
s = standard deviation =    7      
          
Thus,          
          
z = (x - u) * sqrt(n) / s =    -0.05374891      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   -0.05374891   ) =    0.521432393 [ANSWER]

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