Question Find the missing values in the relative frequency table for the concession stand sales data. Solve for variables $a$ , $b$ , $c$ , $d$ , and $e$ to complete the table.

Studdy Solution

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 $a$ , 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.
$0.17 + a + 0.05 = 0.36$

STEP 3

Now, solve for $a$ .
$a = 0.36 - 0.17 - 0.05$

STEP 4

Calculate the value of $a$ .
$a = 0.36 - 0.22$
$a = 0.14$

STEP 5

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

STEP 6

Now, solve for $b$ .
$b = 0.38 - 0.14 - 0.04$

STEP 7

Calculate the value of $b$ .
$b = 0.38 - 0.18$
$b = 0.20$

STEP 8

To find the value of $c$ , 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.
$0.05 + c + 0.02 = Total\, for\, Popcorn$

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.
$Total\, for\, Popcorn = 1 - 0.45 - 0.38$

STEP 10

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

STEP 11

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

STEP 12

Solve for $c$ .
$c = 0.17 - 0.05 - 0.02$

STEP 13

Calculate the value of $c$ .
$c = 0.17 - 0.07$
$c = 0.10$

STEP 14

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

STEP 15

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

STEP 16

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

STEP 17

Now, solve for $e$ .
$e = 0.45 - 0.17 - 0.23$

STEP 18

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

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