Question

Random samples of size \(n\) were selected from binomial populations with population parameters \(p\) given here. Find the mean and the standard deviation of the sampling distribution of the sample proportion \(\hat{p}\) in each case: a. \(n=100, p=.3\) b. \(n=400, p=.1\) c. \(n=250, p=.6\)

Short answer

a. n=100, p=.3 b. n=400, p=.1 c. n=250, p=.6 Answer: a. \(\mu_{\hat{p}}=.3\), \(\sigma_{\hat{p}}\approx0.0458\) b. \(\mu_{\hat{p}}=.1\), \(\sigma_{\hat{p}}\approx0.015\) c. \(\mu_{\hat{p}}=.6\), \(\sigma_{\hat{p}}\approx0.0309\)

Step-by-step solution

Calculate the mean of the binomial distribution

The mean \(µ=np=(100)(.3)=30\)

Calculate the variance of the binomial distribution

The variance \(σ^2=np(1-p)=(100)(.3)(.7)=21\)

Calculate the mean of the sampling distribution of \(\hat{p}\)

The mean of the sampling distribution of \(\hat{p}\) is equal to the population parameter \(p\): \(\mu_{\hat{p}}=p=.3\)

Calculate the standard deviation of the sampling distribution of \(\hat{p}\)

Using the formula \(\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}\), we have: \(\sigma_{\hat{p}}=\sqrt{\frac{.3(1-.3)}{100}}=\sqrt{\frac{.3(.7)}{100}}\approx0.0458\) #b. n=400, p=.1#

Calculate the mean of the binomial distribution

The mean \(µ=np=(400)(.1)=40\)

Calculate the variance of the binomial distribution

The variance \(σ^2=np(1-p)=(400)(.1)(.9)=36\)

Calculate the mean of the sampling distribution of \(\hat{p}\)

The mean of the sampling distribution of \(\hat{p}\) is equal to the population parameter \(p\): \(\mu_{\hat{p}}=p=.1\)

Calculate the standard deviation of the sampling distribution of \(\hat{p}\)

Using the formula \(\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}\), we have: \(\sigma_{\hat{p}}=\sqrt{\frac{.1(1-.1)}{400}}=\sqrt{\frac{.1(.9)}{400}}\approx0.015\) #c. n=250, p=.6#

Calculate the mean of the binomial distribution

The mean \(µ=np=(250)(.6)=150\)

Calculate the variance of the binomial distribution

The variance \(σ^2=np(1-p)=(250)(.6)(.4)=60\)

Calculate the mean of the sampling distribution of \(\hat{p}\)

The mean of the sampling distribution of \(\hat{p}\) is equal to the population parameter \(p\): \(\mu_{\hat{p}}=p=.6\)

Calculate the standard deviation of the sampling distribution of \(\hat{p}\)

Using the formula \(\sigma_{\hat{p}}=\sqrt{\frac{p(1-p)}{n}}\), we have: \(\sigma_{\hat{p}}=\sqrt{\frac{.6(1-.6)}{250}}=\sqrt{\frac{.6(.4)}{250}}\approx0.0309\) In conclusion, the mean and standard deviation of the sampling distribution of the sample proportion \(\hat{p}\) for each case are: a. \(\mu_{\hat{p}}=.3\), \(\sigma_{\hat{p}}\approx0.0458\) b. \(\mu_{\hat{p}}=.1\), \(\sigma_{\hat{p}}\approx0.015\) c. \(\mu_{\hat{p}}=.6\), \(\sigma_{\hat{p}}\approx0.0309\)

Key concepts

Binomial Distribution

A binomial distribution is a specific probability distribution that represents the number of successes in a fixed number of independent and identically distributed Bernoulli trials. In simpler terms, it's like flipping a coin a certain number of times and finding out how many times you get heads. Each trial has two possible outcomes: success or failure. The probability of success is constant for each trial, denoted as \( p \).
A binomial distribution is characterized by its parameters, \( n \) and \( p \), where \( n \) is the total number of trials, and \( p \) is the probability of success in each trial. For example, if you are polling a group of people to find out if they like a particular ice cream flavor, and 30% of the general population does, then \( p = 0.3 \).
Calculating important aspects of the binomial distribution involves different formulas:

• The mean is calculated as: \( \mu = np \)
• The variance is calculated as: \( \sigma^2 = np(1-p) \)

These formulas help us understand the expected number of successes and the variability around this expectation.

Sample Proportion

Sample proportion, denoted as \( \hat{p} \), is an essential concept in statistics when dealing with samples rather than entire populations. It represents the ratio of successes to the total number of trials within the sample. The idea is to use the sample proportion as an estimate of the population proportion, \( p \).
For instance, if you have a sample of 100 people and 30 of them like chocolate ice cream, the sample proportion \( \hat{p} \) is 0.3. This is calculated as the number of successes divided by the sample size: \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes and \( n \) is the total sample size.
In the context of a sampling distribution, the mean of the sample proportion distribution equals the population proportion \( p \).

• The mean of the sample proportion distribution is \( \mu_{\hat{p}} = p \)

This reflects the unbiased nature of the sample proportion as it centers around the true population proportion.

Standard Deviation

Standard deviation is a measure of how spread out numbers are in a distribution. In the context of sampling distribution of the sample proportion \( \hat{p} \), it represents how much the sample proportion is expected to vary from the true population proportion \( p \) across different random samples.
The formula to calculate the standard deviation of the sample proportion distribution is:
\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]where \( p \) is the population proportion and \( n \) is the sample size.
This formula tells us that the variability of \( \hat{p} \) decreases as we increase our sample size. A larger sample size leads to a smaller standard deviation, meaning our estimates of \( p \) become more precise.

• It is vital to understand this concept when estimating population parameters based on samples, as it helps gauge the reliability of the sample's representation of the population.

The precision of the sample proportion can be critical in professions like survey analysis and quality control, where accurate population parameter estimates are crucial.