Question

A coffee company sells bags of coffee beans with an advertised weight of 454 grams. A random sample of 20 bags of coffee beans has an average weight of 457 grams. Weights of coffee beans per bag are known to follow a normal distribution with standard deviation 4 grams.

(a) Construct a 95% confidence interval for the true mean weight of all bags of coffee beans. (Instead of typing ±, simply type +-.)

(b) Provide an interpretation of the confidence interval in (a).

(c) Conduct a hypothesis test at the 5% level of significance to determine if there is evidence that the true mean weight of all bags of coffee beans differs from 454 grams. You will be marked on your calculation of the appropriate test statistic, P-value and a carefully worded conclusion.

(d) Provide an interpretation of the P-value in (c).

(e) Could the confidence interval in (a) have been used to conduct the test in (c)? Explain why or why not. If it could have been used, what would your conclusion be and why? (

Answer 1

a)

Level of Significance ,    α =    0.05          
'   '   '          
z value=   z α/2=   1.96   [Excel formula =NORMSINV(α/2) ]      
                  
Standard Error , SE = σ/√n =   4.0000   / √   20   =   0.8944
margin of error, E=Z*SE =   1.9600   *   0.8944   =   1.7530

concfidnce interval = 457±1.7530

b) there is 95% confidence that true mean lies within confidence interval

c)

Ho :   µ =   454                  
Ha :   µ ╪   454       (Two tail test)          
                          
Level of Significance ,    α =    0.050                  
population std dev ,    σ =    4.0000                  
Sample Size ,   n =    20                  
Sample Mean,    x̅ =   457.0000                  
                          
'   '   '                  
                          
Standard Error , SE = σ/√n =   4.0000   / √    20   =   0.8944      
Z-test statistic= (x̅ - µ )/SE = (   457.000   -   454   ) /    0.8944   =   3.354
                          

p-Value   =   0.0008   [ Excel formula =NORMSDIST(z) ]              
Decision:   p-value<α, Reject null hypothesis                       
Conclusion: There is enough evidence to conclude that true mean weight of all bags of coffee beans differs from 454 gram

d)

the probability of observing the test statistic extreme or more extreme than observed test statistic , when null hypothesis is true is 0.0008

e) 95%   confidence interval is (   455.25   < µ <   458.75   )

since, confidence interval do not contain 454 in interval, so, reject Ho