Lightbulbs of a certain type are advertised as having an average lifetime of 750

Question

Lightbulbs of a certain type are advertised as having an average lifetime of 750 hours. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the true average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected, the lifetime of each bulb was collected. Consequently following table is the statistic of the sample data. Based on given sample statistic (mean and standard deviation), what conclusion would be appropriate for a significance level of 0.05. (Show all hypotheses, test statistic value, P-value, and conclusion in terms of problem context)

Explanation / Answer

Formulating the null and alternative hypotheses,              
              
Ho:   u   >=   750  
Ha:    u   <   750  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical z, as alpha =    0.05   ,      
alpha =    0.05          
zcrit =    -   1.644853627      
              
Getting the test statistic, as              
              
X = sample mean =    738.44          
uo = hypothesized mean =    750          
n = sample size =    50          
s = standard deviation =    38.2          
              
Thus, z = (X - uo) * sqrt(n) / s =    -2.139830992          
              
Also, the p value is              
              
p =    0.016184214          
              
As z < 1.645, and P < 0.05, we   REJECT THE NULL HYPOTHESIS.          

Thus, there is significant evidence that the true average lifetime is smaller that what is advertised. [CONCLUSION]

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