Question

05.BU.A: Two restaurants keep track of their sales per table. At Great Food Restaurant, the average sales per table is $121.65 with a standard deviation of $26.31. At Yummy Restaurant, the average sales per table is $110.83 with a standard deviation of $46.94. You may assume a normal distribution for each restaurant. A large family goes for Sunday dinner at both restaurants on consecutive weeks and coincidentally, their tab at both restaurants is $155.03. (a) Calculate the z-scores to find the relative tab at both restaurants. (b) Relative to the other customers, at which restaurant did the family spend more?These questions focus on Yummy Restaurant. Assume a normal model and use the calculator functions normalcdf and InvNorm. (c) What percent of tables had a tab less than $80.00? (d) What percent of tables had a tab of more than $150.00? (e) What percent of tables had a tab between $110.00 and $140.00? (f) The “big spenders” spend in the top 10% of diners. How much does one need to spend to be considered a :big spender? (g) In actual dollars, what is the IQR for tab? (Hint: Between what two percents is the IQR?

Answer 1

a)for great food restaurant

µ=   121.65
σ=   26.31
X=   155.03
Z=(X-µ)/σ=   (155.03-121.65)/26.31)=       1.2687
for yummy restaurant

µ=   110.83
σ=   46.94
X=   155.03
Z=(X-µ)/σ=   (155.03-110.83)/46.94)=       0.9416

b)

family spend more on great food restaurant

.............

c)

µ =    110.83      
σ =    46.94      
          
P( X ≤    80   ) = P( (X-µ)/σ ≤ (80-110.83) /46.94)  
=P(Z ≤   -0.657   ) =   0.2557 = 25.57% excel formula for probability from z score is =NORMSDIST(Z)

......

d)

µ =    110.83                  
σ =    46.94                  
                      
P ( X ≥   150.00   ) = P( (X-µ)/σ ≥ (150-110.83) / 46.94)              
= P(Z ≥   0.834   ) = P( Z <   -0.834   ) =    0.2020 = 20.20% (answer)
excel formula for probability from z score is =NORMSDIST(Z)                  
..............

e)

µ =    110.83                              
σ =    46.94                              
we need to calculate probability for ,                                  
P (   110   < X <   140   )                  
=P( (110-110.83)/46.94 < (X-µ)/σ < (140-110.83)/46.94 )                                  
                                  
P (    -0.018   < Z <    0.621   )                   
= P ( Z <    0.621   ) - P ( Z <   -0.018   ) =    0.7328   -    0.4929   =    0.2399 = 23.99%
excel formula for probability from z score is =NORMSDIST(Z)

.............

f)

µ=   110.83                  
σ =    46.94                  
proportion=   0.9                  
                      
Z value at    0.9   =   1.28   (excel formula =NORMSINV(   0.9   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   1.28   *   46.94   +   110.83  
X   =   170.99   (answer)          
one need to spend to 170.99 ~171 be considered a big spender

thanks

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