The age of the children in kindergarten on the first day of school is uniformly distributed between 4.8 and 5.8 years old. A first time kindergarten child is selected at random. Round all answers to two decimal places.

1. The mean of the distribution is

a. 5.4

b. 5.2

c. 5.3

d. 5.5

2. The standard deviation is

a. 0.194

b. 0.289

c. 0.333

d. 0.261

3. The probability that the child will be older than 5 years old is

a. 0.3

b. 0.8

c. 0.2

d. 0.7

4. The probability that the child will be between 5.2 and 5.7 years old is.

a. 0.6

b. 0.7

c. 0.8

d. 0.5

5. If such a child is at the 45th percentile, how old is that child?

a. 5.25

b. 5

c. 4.75

d. 5.5

(1) (c) 5.30 years.

(2) (b) 0.289.

(3) (b) 0.80.

(4) (d) 0.50.

(5) (a) 5.25 years.

Step-by-step explanation:

Let X = age of the children in kindergarten on the first day of school.

The random variable X follows a continuous Uniform distribution with parameters a = 4.8 years and b = 5.8 years.

The probability density function function of X is:

(1)

The expected value of a Uniform random variable is:

Compute the mean of X as follows:

Thus, the  mean of the distribution is (c) 5.30 years.

(2)

The standard deviation of a Uniform random variable is:

Compute the standard deviation of X as follows:

Thus, the standard deviation of the distribution is (b) 0.289.

(3)

Compute the probability that a randomly selected child is older than 5 years old as follows:

Thus, the probability that a randomly selected child is older than 5 years old is (b) 0.80.

(4)

Compute the probability that a randomly selected child is between 5.2 years and 5.7 years old as follows:

Thus, the probability that a randomly selected child is between 5.2 years and 5.7 years old is (d) 0.50.

(5)

It is provided that a randomly selected child is at the 45th percentile.

This implies that:

P (X < x) = 0.45

Compute the value of x as follows:

Thus, the age of the child at the 45th percentile is (a) 5.25 years.

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