Chapter 7.1, Problem 16E

(a)

To determine

The description about the population distribution.

Answer to Problem 16E

The population distribution is approximately a binomial distribution with p = 0.75.

Explanation of Solution

Given information:

Number of white beads = 1000

Number of red beads = 3000

Number of beads for SRS = 50

Mean of the Normal distribution = 0.75

Standard deviation = 0.06

To find the total population, add the number of white beads and red beads that is, 1000 beads and 3000 beads.

  $1000+3000=4000$

Find the proportion of red beads by dividing 3000 by 4000.

  $\frac{3000}{4000}=0.75$

The above population distribution is approximately a binomial distribution with $p=0.75$ .

Hence, the population distribution is approximately a binomial distribution with p = 0.75.

(b)

To determine

The description about the sampling distribution of $\overline{p}$ and the difference between the population distribution and the sampling distribution.

Explanation of Solution

Given information:

Number of white beads = 1000

Number of red beads = 3000

Number of beads for SRS = 50

Mean of the Normal distribution = 0.75

Standard deviation = 0.06

The values of sample proportion $\widehat{p}$ follow the Normal distribution whose mean is 0.75 and the standard deviation is 0.06.

This means that the sampling distribution of $\widehat{p}$ is the Normal distribution with mean of 0.75 minutes and standard deviation of 0.06.

Also, in the given data the sampling distribution of $\overline{x}$ describes the proportion of red beads in a sample size of 50.

The sampling distribution differs from the population distribution by its Normal distribution.