Q.16 - The Practice of Statistics for AP

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? 

Step-by-Step Solution

Answer

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

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

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).