Please work with the two cases below about data distribution with explanation.

I. In an experiment on the behavior of young children, each child is placed in an area

with five toys. The response of interest is the number of toys that the child plays with.

Past experiments with many subjects have shown that the probability distribution of the

number of toys played with is as follows:

Number of toys :0 1 2 3

Probability : 0.05 0.20 0.44 0.31

Now assume that we independently sample 14 additional children (that gives 15 children

total in the sample) and then look only at whether or not they play with any toys. Let

Y be the number of children in our sample who play with at least 1 toy.

1) What is the distribution of Y and what are the parameters of that distribution?

2) What are the expected value and the variance of Y ?

II. Achievement test scores of all high school seniors in a Canadian province have mean

60 and standard deviation 8. A random sample of n = 100 students from one large high

school in this province had a mean score of 58.

(a) Describe the approximate distribution of the sample mean achievement test scores of

high school seniors.

(b) Describe how the distribution of the sample mean achievement test scores would

change if a larger sample of students were taken.

(c) Is there any evidence to suggest that this high school is inferior? Test at a significance

level of = 0.05.