Solved on Dec 18, 2023

Find the missing values in the relative frequency table for the concession stand sales data. Solve for variables aa, bb, cc, dd, and ee to complete the table.

STEP 1

Assumptions
1. Each cell in the relative frequency table represents the proportion of total sales for a particular combination of concession stand item and beverage.
2. The 'Total' row and column represent the sum of the relative frequencies for that row or column.
3. The sum of all relative frequencies in the table should equal 1, as they represent the whole of the concession stand sales.

STEP 2

To find the value of aa, we need to use the total for the 'Soda' row. We know that the sum of the relative frequencies in this row is 0.36.
0.17+a+0.05=0.360.17 + a + 0.05 = 0.36

STEP 3

Now, solve for aa.
a=0.360.170.05a = 0.36 - 0.17 - 0.05

STEP 4

Calculate the value of aa.
a=0.360.22a = 0.36 - 0.22
a=0.14a = 0.14

STEP 5

To find the value of bb, we look at the 'Total' column for 'Pizza'. The sum of the relative frequencies for 'Pizza' is 0.38.
0.14+b+0.04=0.380.14 + b + 0.04 = 0.38

STEP 6

Now, solve for bb.
b=0.380.140.04b = 0.38 - 0.14 - 0.04

STEP 7

Calculate the value of bb.
b=0.380.18b = 0.38 - 0.18
b=0.20b = 0.20

STEP 8

To find the value of cc, we use the 'Water' row. We don't have the total for this row, but we can use the 'Total' column for 'Popcorn' to find it. We know that the sum of the relative frequencies for 'Popcorn' plus the known value for 'Nodrink' row (0.02) and 'Soda' row (0.05) should equal the total for the 'Popcorn' column.
0.05+c+0.02=TotalforPopcorn0.05 + c + 0.02 = Total\, for\, Popcorn

STEP 9

Since we don't have the total for 'Popcorn', we can find it by subtracting the known totals for 'Hot Dog' and 'Pizza' from 1.
TotalforPopcorn=10.450.38Total\, for\, Popcorn = 1 - 0.45 - 0.38

STEP 10

Calculate the total for 'Popcorn'.
TotalforPopcorn=10.83Total\, for\, Popcorn = 1 - 0.83
TotalforPopcorn=0.17Total\, for\, Popcorn = 0.17

STEP 11

Now we can solve for cc.
0.05+c+0.02=0.170.05 + c + 0.02 = 0.17

STEP 12

Solve for cc.
c=0.170.050.02c = 0.17 - 0.05 - 0.02

STEP 13

Calculate the value of cc.
c=0.170.07c = 0.17 - 0.07
c=0.10c = 0.10

STEP 14

To find the value of dd, we use the 'Water' row. We now know the value of cc, so we can use the sum of the relative frequencies for 'Water'.
0.23+0.20+0.10=d0.23 + 0.20 + 0.10 = d

STEP 15

Calculate the value of dd.
d=0.23+0.20+0.10d = 0.23 + 0.20 + 0.10
d=0.53d = 0.53

STEP 16

Finally, to find the value of ee, we use the 'Total' column for 'Hot Dog'. The sum of the relative frequencies for 'Hot Dog' is 0.45.
0.17+0.23+e=0.450.17 + 0.23 + e = 0.45

STEP 17

Now, solve for ee.
e=0.450.170.23e = 0.45 - 0.17 - 0.23

STEP 18

Calculate the value of ee.
e=0.450.40e = 0.45 - 0.40
e=0.05e = 0.05
The missing values in the relative frequency table are as follows:
a=0.14b=0.20c=0.10d=0.53e=0.05 \begin{array}{l} a = 0.14 \\ b = 0.20 \\ c = 0.10 \\ d = 0.53 \\ e = 0.05 \\ \end{array}

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