A company wants to determine whether its consumer product ratings (0-10) have ch

Question

A company wants to determine whether its consumer product ratings (0-10) have changed from last year to tliis year. The table shows the company's product ratings from the same eight consumers for last year and this year. At alpha = 0.05, is there enough evidence to conclude that the product ratings have changed? Define the difference d = Rating last year - Rating this year. Assume that the distributioin of difference of ratings is approximately normal. Find the sample mean and standard deviation of the differences. State the null and alternative hypotheses to test whether the ratings have changed. Find the value of the standardized test statistic. Determine the critical value(s) and the rejection regions. Make a decision and interpret it.

Explanation / Answer

a)

The differences are

0
-2
-2
-3
0
1
-1
-1
  

Calculating the mean of the differences (third column):              
              
XD =    -1   [ANSWER, MEAN]      

              
Calculating the standard deviation of the differences (third column):              
              
s =    1.154700538   [ANSWER, STANDARD DEVIATION]

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b)      

Formulating the null and alternative hypotheses,              
              
Ho:   ud   =   0  
Ha:   ud   =/   0   [ANSWER]

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c)

At level of significance =    0.05          

As we can see, this is a    two   tailed test.  

              
Thus, the standard error of the difference is sD = s/sqrt(n):              
              
sD =    0.40824829          
              
              
As t = [XD - uD]/sD, where uD = the hypothesized difference =    0   , then      
              
t =    -2.449489743   [ANSWER, TEST STATISTIC]

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d)      
              
As df = n - 1 =    7          
              
Then the critical value of t is              
              
tcrit =    +/-   2.364624252   [ANSWER]

Hence, we reject Ho when t < -2.365 or t > 2.365. [ANSWER]

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e)
              
As |t| > 2.365, we REJECT HO.

There is significant evidence that the product ratings have changed from last year to this year at 0.05 level. [CONCLUSION]

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