Normal Approximation to Binomial and Sampling Distributions

Flashcards reviewing normal approximation to the binomial distribution, continuity corrections, sampling distributions, standard error, and the Central Limit Theorem.

Under what condition can a normal distribution be used to approximate a binomial distribution?

When the number of trials is large enough such that both $n \times p > 10$ and $n \times (1 - p) > 10$.

When approximating a binomial distribution with a normal distribution, what are the parameters for the mean and standard deviation?

The mean is $\mu = n \times p$ and the standard deviation is $\sigma = \sqrt{n \times p \times (1 - p)}$ (or $\sqrt{n \times p \times q}$).

Why is a continuity correction necessary when approximating a binomial distribution with a normal distribution?

Because the binomial distribution is discrete while the normal distribution is continuous, requiring a $0.5$ adjustment so the continuous curve properly covers the discrete histogram intervals.

How is the discrete binomial probability $P(49 \le X \le 51)$ adjusted using continuity correction for normal approximation?

It is adjusted to $P(48.5 < X < 51.5)$ by subtracting $0.5$ from the lower bound and adding $0.5$ to the upper bound.

In the salmon example with $175$ trials and success probability $0.3$, how are the normal approximation threshold conditions verified?

$n \times p = 175 \times 0.3 = 52.5$ and $n \times (1 - p) = 175 \times 0.7 = 122.5$, both of which are greater than $10$.

In R, what code expression calculates the probability of between 50 and 75 salmon spawning using normal approximation with continuity correction?

pnorm(75.5, mean, sd) - pnorm(49.5, mean, sd).

What is a sampling distribution?

The distribution of a sample statistic (such as a sample mean, sample standard deviation, or sample proportion) obtained by repeatedly taking samples from a population.

What is the relationship between the mean of the sampling distribution of the sample mean ($\mu_{\bar{X}}$) and the population mean ($\mu$)?

The mean of the sample means is equal to the population mean ($\mu_{\bar{X}} = \mu$).

What is the formula for the standard error of the mean?

$\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size.

What happens to the standard error of the sample mean as the sample size $n$ increases?

The standard error becomes smaller, making the distribution of the sample mean narrower, less uncertain, and closer to the population parameter.

According to the Central Limit Theorem, what sample size condition ensures that the sample mean follows a normal distribution regardless of the population distribution shape?

A sample size greater than 30 ($n > 30$).

Under the Central Limit Theorem, what condition must be met for the sample mean to follow a normal distribution if the sample size is less than 30 ($n < 30$)?

The original population distribution itself must follow a normal distribution.