The local bakery bakes more than a thousand 1-pound loaves of bread daily, and the weights of these loaves varies. The mean weight is 1.6 lb. and 1 oz., or 754 grams. Assume the standard deviation of the weights is 30 grams and a sample of 42 loaves is to be randomly selected. (a) Find the mean of this sampling distribution. (Give your answer correct to nearest whole number.) Incorrect: Your answer is incorrect. grams (b) Find the standard error of this sampling distribution. (Give your answer correct to two decimal places.) (c) What is the probability that this sample mean will be between 747 and 761? (Give your answer correct to four decimal places.) (d) What is the probability that the sample mean will have a value less than 752? (Give your answer correct to four decimal places.) (e) What is the probability that the sample mean will be within 7 grams of the mean?
Answer)
As the data is normally distributed we can use standard normal z table to estimate the answers
Z = (x-mean)/(s.d/√n)
Given mean = 754
S.d = 30
A)
Mean js same as population mean = 754
B)
Standard error = s.d/√n = 30/√42 = 4.63
C)
P(747<x<761) = p(x<761) - p(x<747)
P(x<761)
Z = (761 - 754)/4.63 = 1.51
From z table, P(z<1.51) = 0.9345
P(x<747)
Z = (747 - 754)/4.63 = -1.51
From z table, P(z<-1.51) = 0.0655
Required probability is 0.9345 - 0.0655 = 0.869
D)
P(x<752)
Z = (752 - 754)/4.63 = -0.43
From z table, P(z<-0.43) = 0.3336
E)
0.869 (calculated in part c)
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