Chapter 5: Normal Probability Distributions

Comprehensive practice flashcards covering Chapter 5: Normal Probability Distributions, including standard normal distributions, R commands, percentiles, sampling distributions, and the Central Limit Theorem.

What is a normal distribution?

A continuous probability distribution for a random variable $x$, whose graph is called the normal curve.

What are the four main properties of a normal distribution?

1. The mean, median, and mode are equal.

2. The curve is bell-shaped and symmetric about the mean.

3. The total area under the curve equals $1$.

4. The curve approaches but never touches the x-axis as it extends away from the mean.

5. Between μ − σ and μ + σ at the center of the curve, the
   graph curves downward. It curves upward to the left of μ − σ and to the right of μ + σ The points at which the curve changes from upward to downward are called inflection points.

Where do inflection points occur on a normal curve, and what are they?

Inflection points are the points where the curve changes from curving upward to curving downward. They occur at $\mu - \sigma$ and $\mu + \sigma$.

How do the parameters $\mu$ and $\sigma$ affect the shape and location of a normal curve?

The mean $\mu$ gives the line of symmetry, and the standard deviation $\sigma$ describes how much the data is spread out.

In this comparison of normal distributions, what parameters correspond to the black, red, and blue curves?

The black curve has $\mu = 25$ and $\sigma = 3$, the red curve has $\mu = 35$ and $\sigma = 3$, and the blue curve has $\mu = 25$ and $\sigma = 6$.

What distribution is represented by this graph with a center at $0$ and a horizontal axis typically spanning from $-3$ to $3$?

The standard normal distribution, which has a mean of $\mu = 0$ and a standard deviation of $\sigma = 1$.

What formula is used to transform a data value $x$ into a z-score?

$$z = \frac{x - \mu}{\sigma}$$

What is the R command to find the area to the left of a given z-score under the standard normal curve?

pnorm(zscore, mean=0, sd=1, lower.tail=T)

What are the two ways to write the R command to find the area to the right of a z-score under the standard normal curve?

1-pnorm(zscore, mean=0, sd=1, lower.tail=T) OR pnorm(zscore, mean=0, sd=1, lower.tail=F)

How can you calculate normal distribution probabilities in R directly from a data value without first converting it to a z-score?

By including the mean and standard deviation directly in the command: pnorm(datavalue, mean=\mu, sd=\sigma, lower.tail=T or F)

Which R command is used to invert pnorm() and find the z-score or data point associated with a given area or probability?

qnorm()

What formula converts a z-score back into a raw data value $x$?

$$x = \mu + z\sigma$$

What is a sampling distribution?

The probability distribution of a sample statistic that is formed when samples of size $n$ are repeatedly taken from a population.

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

$$\mu_{\bar{x}} = \mu$$

What is the standard error of the mean, and what is its formula?

The standard deviation of the sampling distribution of sample means, given by: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

According to the Central Limit Theorem, under what sample size condition does the sampling distribution of sample means approximate a normal distribution for any population?

When samples of size $n \ge 30$ are drawn.

According to the Central Limit Theorem, what sample size $n$ is required for the sampling distribution of sample means to be normally distributed if the population itself is normally distributed?

Any sample size $n$.

In the commuting distance example, what does this symmetric, bell-shaped distribution represent compared to the skewed population distribution?

The sampling distribution of sample means for random samples of size $n = 30$, centered at $\mu_{\bar{x}} = 16$.

What is the z-score formula used when finding probabilities for a sample mean $\bar{x}$?

$$z = \frac{\bar{x} - \mu_{\bar{x}}}{\sigma_{\bar{x}}} = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$$