89 THE VIDEO GAME SATISFACTION RATING CASE OsVideoGame The m

8.9 THE VIDEO GAME SATISFACTION RATING CASE OsVideoGame The mean of the sample of 65 customer satisfaction ratings in Table 1.7 is 42.95. If we let u denote the mean of all possible customer satisfaction ratings for the XYZ Box video game system, and as- sume that the population standard deviation equals 2.64: a Calculate 95 percent and 99 percent confidence intervals for Au. b Using the 95 percent confidence interval, can we be 95 percent confident that wis at least 42 (recall that a very satisfied customer gives a rating of at least 42)? Explain. c Using the 99 percent confidence interval, can we be 99 percent confident that Au is at least 42? Explain. d Based on your answers to parts b and c, how convinced are you that the mean satisfaction rating is at least 42?

Solution

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
              
X = sample mean =    42.95          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    2.64          
n = sample size =    65          
              
Thus,              
              
Lower bound =    42.30820646          
Upper bound =    43.59179354          
              
Thus, the confidence interval is              
              
(   42.30820646   ,   43.59179354   ) [95% confidence interval]

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Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
              
X = sample mean =    42.95          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    2.64          
n = sample size =    65          
              
Thus,              
              
Lower bound =    42.10654032          
Upper bound =    43.79345968          
              
Thus, the confidence interval is              
              
(   42.10654032   ,   43.79345968   ) [99% confidence interval]

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b.

Yes, because even the lower bound is higher than 42.

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c.

Also yes, because even the lower bound is higher than 42.

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d. We are convinced at 0.025 and 0.005 level; the right tails of these confidence intervals.