Weather forecasters would like to report on temperatures in Washington, DC. In p

Question

Weather forecasters would like to report on temperatures in Washington, DC. In particular, they would like to compare the temperatures in December to temperatures in January to help potential tourists understand whether there is a difference. They believe it is colder in January but want to conduct a test to support their assertion. The average December temperature in a random sample of 16 days is 35 and the average January temperature in a random sample of 20 days is 33. The standard deviations based on historic information are 4 for December and 5 for January. Specify the competing hypothesis to determine if it is colder in January than in December. Find the critical value(s) of the appropriate test at the 5% significance level. Calculate the value of the test statistic. Make a conclusion at the 5% significance level.

Explanation / Answer

a)

Let

u1 = December mean temperature
u2 = January mean temperature

Formulating the null and alternative hypotheses,              
              
Ho:   u1 - u2   <=   0  
Ha:   u1 - u2   >   0   [ANSWER]

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b)

Here,

n1 = sample size of group 1 =    16          
n2 = sample size of group 2 =    20          

Thus, df = n1 + n2 - 2 =    34          

Now, the critical value for t is, at 0.05 right tailed,   by table/technology,          
              
tcrit =        1.690924255   [ANSWER]

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c)
  
Calculating the means of each group,              
              
X1 =    35          
X2 =    33          
              
Calculating the standard deviations of each group,              
              
s1 =    4          
s2 =    5          
              
Thus, the standard error of their difference is, by using sD = sqrt(s1^2/n1 + s2^2/n2):              
              
n1 = sample size of group 1 =    16          
n2 = sample size of group 2 =    20          

Also, sD =    1.5          
              
Thus, the t statistic will be              
              
t = [X1 - X2 - uD]/sD =    1.333333333   [ANSWER, TEST STATISTIC]

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d)

As t < 1.691, we fail to reject Ho.

There is no significant evidence that it is colder on the average in January than in December at 0.05 level. [CONCLUSION]      
              

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