Question

Scooping beads A statistics teacher fills a large container with $1000$white and $3000$red beads and then mixes the beads thoroughly. She then has her students take repeated SRSs of $50$beads from the container. After many SRSs, the values of the sample proportion pˆ of red beads are approximated well by a Normal distribution with mean of $0.75$and standard deviation of $0.06$.

(a) What is the population? Describe the population distribution. (b) Describe the sampling distribution of $pˆ$. How is it different from the population distribution?

Short answer

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

b). Proportion of red beads in a sample of size $50$.

Step-by-step solution

Part (a) Step 1: Given Information 

The values of the sample proportion $pˆ$ of red beads are approximated well by a Normal distribution with mean of $0.75$ and standard deviation of $0.06$.

Part (a) Step 2: Explanation 

Individuals who are being examined make up the population.

A huge container contains POPULATION $=4000$beads, $1000$of which are white and $3000$of which are red.

The population distribution is approximately a binomial distribution with $p=\frac{3000}{4000}=0.75$(proportion of red beads).

Part (b) Step 1: Given Information 

The values of the sample proportion $pˆ$of red beads are approximated well by a Normal distribution with mean of $0.75$ and standard deviation of $0.06$.

Part (b) Step 2: Explanation 

The sample proportion $\hat{p}$values are distributed normally, with a mean of $0.75$and a standard deviation of $0.06$.

As a result $\hat{p}$'s sampling distribution is the Normal distribution, with a mean of $0.75$and a standard deviation of $0.06$.

The proportion of red beads in a sample of size $50$is described by the sampling distribution of $\overline{x}$Its difference distribution distinguishes it from the population distribution (Normal instead of binomial).