Question

The weights of boxes of crackers produced by a certain manufacturer have a Normal distribution with a mean of 255 grams and a standard deviation of 3.5 grams.

(a) (4 points) What percent of boxes of crackers weigh more than 260 grams?

(b) (4 points) What is the z-score for a box of crackers that weighs 260 grams?

(c) (4 points) What percent of observations from a standard Normal distribution N(0, 1) take values larger than the standardized value from part (b)? (

d) (2 points) Are your answers to parts (a) and (c) similar? (Don’t simply answer yes or no. Explain how they are or are not similar.)

(e) (4 points) What is the weight that should be stamped on each box so that only 1% of boxes are underweight? need asap thank you

Answer 1

a)

µ =    255                  
σ =    3.5                  
                      
P ( X > 260   ) = P( (X-µ)/σ ≥ (260-255) / 3.5)              
= P(Z > 1.429   ) = P( Z <   -1.429   ) =    0.0766 or 7.66% (answer)

b)

Z=(X-µ)/σ=   (260-255)/3.5)=       1.429

c) P(Z > 1.429   ) = P( Z ≤ -1.429   ) =    0.0766 or 7.66% (answer)

d) yes,

Both values are same,as original distribution is converted to standard normal distribution and it is same as the original distribution.

e)

µ=   255                  
σ =    3.5                  
proportion=   0.01                  
                      
Z value at    0.01   =   -2.326   (excel formula =NORMSINV(   0.01   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   -2.326   *   3.5   +   255  
X   =   246.86   (answer)