Question

According to the Current Results website, the state of California has a mean annual rainfall of 23 inches, whereas the state of New York has a mean annual rainfall of 45 inches. Assume that the standard deviation for both states is 3 inches. A sample of 30 years of rainfall for California and a sample of 45 years of rainfall for New York has been taken. Use z-table. a. Show the probability distribution of the sample mean annual rainfall for California. This is a graph of a normal distribution with E(-/x) and 0x (to 4 decimals). b. What is the probability that the sample mean is within 1 inch of the population mean for California? (to 4 decimals) c. What is the probability that the sample mean is within 1 inch of the population mean for New York? (to 4 decimals) d. In which case, part (b) or part (c), is the probability of obtaining a sample mean within 1 inch of the population mean greater? Why? The probability of being within inch is for New York in part (c) because the sample size is

Answer 1

a)

Mean = 23 (from which 50% values fall left and 50% values fall right).

b)

µ =    23                                  
σ =    3                                  
n=   30                                  
we need to calculate probability for ,                                      
22   ≤ X ≤    24                              
X1 =    22   ,    X2 =   24                      
                                      
Z1 =   (X1 - µ )/(σ/√n) = (   22   -   23   ) / (   3   / √   30   ) =   -1.83
Z2 =   (X2 - µ )/(σ/√n) = (   24   -   23   ) / (   3   / √   30   ) =   1.83
                                      
P (   22   < X <    24   ) =    P (    -1.8   < Z <    1.8   )   
                                      
= P ( Z <    1.83   ) - P ( Z <   -1.83   ) =    0.96606   -    0.03394   =    0.9321  
excel formula for probability from z score is =NORMSDIST(Z)                                      

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c)

µ =    45                                  
σ =    3                                  
n=   45                                  
we need to calculate probability for ,                                      
44   ≤ X ≤    46                              
X1 =    44   ,    X2 =   46                      
                                      
Z1 =   (X1 - µ )/(σ/√n) = (   44   -   45   ) / (   3   / √   45   ) =   -2.24
Z2 =   (X2 - µ )/(σ/√n) = (   46   -   45   ) / (   3   / √   45   ) =   2.24
                                      
P (   44   < X <    46   ) =    P (    -2.2   < Z <    2.2   )   
                                      
= P ( Z <    2.24   ) - P ( Z <   -2.24   ) =    0.98733   -    0.01267   =    0.9747  
excel formula for probability from z score is =NORMSDIST(Z)                                      

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d)

The probability of being within inch is greater for New York in part (c) because the sample size is larger for Newyork

Please let me know in case of any question.

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