An amusement park, Dino World, considering adding some new attractions, conducted a study over several typical days and found that, of 10,000 families entering the park, 1020 brought just one child (defined as younger than age 12), 3370 brought two children, 3510 brought three children, 1340 brought four children, 510 brought five children, 80 brought six children and 170 brought no children at all.
a) Find the expected number of children per family attending the park.
b) If the expected number is greater than 2.9 they will add more attractions. Will then add more?
a) To find the expected number of children per family attending the park, we need to calculate the average number of children per family.
First, let's calculate the total number of families:
Number of families with one child: 1020
Number of families with two children: 3370
Number of families with three children: 3510
Number of families with four children: 1340
Number of families with five children: 510
Number of families with six children: 80
Number of families with no children: 170
Total number of families = 1020 + 3370 + 3510 + 1340 + 510 + 80 + 170 = 10000
Next, let's calculate the total number of children:
Number of children from families with one child: 1020 x 1 = 1020
Number of children from families with two children: 3370 x 2 = 6740
Number of children from families with three children: 3510 x 3 = 10530
Number of children from families with four children: 1340 x 4 = 5360
Number of children from families with five children: 510 x 5 = 2550
Number of children from families with six children: 80 x 6 = 480
Number of children from families with no children: 170 x 0 = 0
Total number of children = 1020 + 6740 + 10530 + 5360 + 2550 + 480 + 0 = 36280
To find the expected number of children per family, we divide the total number of children by the total number of families:
Expected number of children per family = Total number of children / Total number of families
Expected number of children per family = 36280 / 10000 = 3.628
Therefore, the expected number of children per family attending the park is approximately 3.628.
b) If the expected number of children per family is greater than 2.9, they will add more attractions. In this case, the expected number of children per family is 3.628, which is greater than 2.9. Therefore, they will add more attractions.
To find the expected number of children per family attending the park, you need to calculate the weighted average of the number of children per family.
a) To calculate the expected number of children per family:
First, you need to determine the total number of families in the sample. You can do this by summing up the number of families in each category. From the given data, the number of families is as follows:
Number of families with one child: 1020
Number of families with two children: 3370
Number of families with three children: 3510
Number of families with four children: 1340
Number of families with five children: 510
Number of families with six children: 80
Number of families with no children: 170
Total number of families = 1020 + 3370 + 3510 +
1340 + 510 + 80 + 170
= 10,000
Next, you need to calculate the weighted average. Multiply the number of children by the corresponding number of families, then sum up these products, and divide by the total number of families.
Expected number of children = [(1020 x 1) + (3370 x 2) + (3510 x 3) +
(1340 x 4) + (510 x 5) + (80 x 6) + 0] / 10,000
= 10,000/10,000
= 1.0
Therefore, the expected number of children per family attending the park is 1.0.
b) Since the expected number of children per family is 1.0, which is less than 2.9, the amusement park should add more attractions.
In conclusion, based on the given data, the expected number of children per family attending the park is 1.0, and the amusement park should consider adding more attractions.