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.

2.668

To find the expected number of children per family attending the park, we need to calculate the average number of children per family.

We can do this by first calculating the total number of children and then dividing it by the total number of families.

Total number of children = (Number of families with 1 child x 1) + (Number of families with 2 children x 2) + (Number of families with 3 children x 3) + (Number of families with 4 children x 4) + (Number of families with 5 children x 5) + (Number of families with 6 children x 6)

= (1020 x 1) + (3370 x 2) + (3510 x 3) + (1340 x 4) + (510 x 5) + (80 x 6)

= 1020 + 6740 + 10530 + 5360 + 2550 + 480

= 35980

Next, we calculate the total number of families by adding the number of families with children to the number of families without children.

Total number of families = (Number of families with 1 child) + (Number of families with 2 children) + (Number of families with 3 children) + (Number of families with 4 children) + (Number of families with 5 children) + (Number of families with 6 children) + (Number of families with no children)

= 1020 + 3370 + 3510 + 1340 + 510 + 80 + 170

= 10000

Finally, we calculate the expected number of children per family by dividing 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

= 35980 / 10000

= 3.598

Therefore, the expected number of children per family attending the park is approximately 3.598.