Question

A random sample of size \(n=50\) is selected from a binomial distribution with population proportion \(p=.7\) a. What will be the approximate shape of the sampling distribution of \(\hat{p} ?\) b. What will be the mean and standard deviation (or standard error) of the sampling distribution of \(\hat{p} ?\) c. Find the probability that the sample proportion \(\hat{p}\) is less than 8 .

Short answer

Answer: The approximate shape of the sampling distribution of \(\hat{p}\) is a normal distribution with a mean of 0.7 and a standard deviation (standard error) of approximately 0.0648. The probability that the sample proportion \(\hat{p}\) is less than 8 is 1 (or 100%), as the value of \(\hat{p}\) should be within the range of 0 to 1, and 8 is outside this range.

Step-by-step solution

Identify the Central Limit Theorem conditions

The Central Limit Theorem (CLT) conditions are: 1. The random variable is the sum of a large number of independent and identically distributed variables. 2. The sample size is large (typically n ≥ 30). In our case, we have a binomial distribution with a sample size (\(n=50\)) larger than 30. Therefore, the CLT conditions are met.

Approximate the sampling distribution of \(\hat{p}\)

Since the CLT conditions are met, we can approximate the sampling distribution of \(\hat{p}\) with a normal distribution. The mean and standard deviation of the binomial distribution can be used to describe the shape of the sampling distribution.

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

Using the formulas for the mean and standard deviation of a binomial distribution, we can calculate the mean and standard deviation of the sampling distribution of \(\hat{p}\). Mean: \(\mu_{\hat{p}} = p = 0.7\) Standard deviation (standard error): \(\sigma_{\hat{p}} = \sqrt{\dfrac{p(1-p)}{n}}\) Plug in the values: \(\sigma_{\hat{p}} = \sqrt{\dfrac{0.7(1-0.7)}{50}}\) \(\sigma_{\hat{p}} \approx 0.0648\)

Find the probability that the sample proportion \(\hat{p}\) is less than 8

The probability that the sample proportion is less than 8 is 1 (or 100%), since we know that the value of \(\hat{p}\) should be within the range of 0 to 1, and 8 is outside this range.

Key concepts

Binomial Distribution

The binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states. For example, these states could be "success" or "failure." This is why it's often used in scenarios where there are a fixed number of trials or experiments. Each trial is independent of others, and the probability of success is the same for each trial. In this case, with a proportion parameter (\( p \)) of 0.7, we are looking at situations where there is a 70% chance of success on each trial. Some key points about binomial distributions include:

• They are defined by two parameters: the number of trials (\( n \)) and the probability of success (\( p \)).
• Each trial is independent.
• Useful for modeling binary problems, like yes/no surveys and pass/fail tests.

The mean (\( \mu \)) of a binomial distribution is given by the formula \( n \times p \), and the standard deviation (\( \sigma \)) is calculated as \( \sqrt{n \times p \times (1-p)}\). These are used to approximate the sampling distribution when the sample size is large.

Sampling Distribution

A sampling distribution refers to the probability distribution of a statistic obtained by selecting random samples from a population. The most commonly discussed example is the sampling distribution of the sample mean.When working with proportions, as in our problem, we often talk about the sampling distribution of \( \hat{p} \), which represents the sample proportion.Due to the Central Limit Theorem, when the sample size is sufficiently large (typically \( n \geq 30 \)), the sampling distribution of the sample proportion \( \hat{p} \) becomes approximately normal. This is because the means of these samples will stack around the true population proportion (\( p \)) creating a bell-shaped curve.The approximate normality of \( \hat{p} \) comes in handy as it simplifies calculations like finding probabilities. This is particularly useful in hypothesis testing and creating confidence intervals.

Standard Error

Standard Error (SE) is crucial in understanding the variability of a sample statistic. It indicates how much the sample proportion is expected to vary from the actual population proportion. In this context, Standard Error applies to the sampling distribution of the sample proportion (\( \hat{p} \)). It's calculated by:\[\sigma_{\hat{p} } = \sqrt{\frac{p(1-p)}{n}}\]This formula reflects that the SE decreases as the sample size (\( n \)) increases — meaning more data typically gives a more accurate estimate. In our problem, using \( p = 0.7 \) and \( n = 50 \), we find the SE to be approximately 0.0648. This quantifies the expected spread of sample proportions around the true population proportion of 0.7.Understanding Standard Error helps in determining confidence in statistical estimates and predictions, which is fundamental in statistical decision-making.