MATH 1680 Chapter 7

Since the probability of observing a specific value of a continuous random variable is 0, the following probabilities are equivalent:

P(a < X < b) = P(a <_ X < b) = P(a < X <_ b) = P(a <_ X <_ b)

Values of normal random variables and their z-scores are ___________ related (x=μ+zσ), so a plot of observations of normal variables against their expected z-scores will be linear.

linearly

What values of x are associated with the inflection points of a normal curve?

mean + standard deviation and mean - standard deviation

The idea behind finding the expected z-score is, if the data come from a ___________________, we could predict the area to the left of each data value.

normally distributed population

What value of x is associated with the peak of a normal curve?

the mean

If the possible values of a uniform density function go from 0 through n, what is the height of the rectangle?

1/n

What does it mean to say that a continuous random variable is normally distributed?

A continuous random variable is normally distributed or has a normal probability distribution, if its relative frequency histogram has the shape of a normal curve

What is a normal probability plot?

A normal probability plot is a graph that plots observed data versus normal scores.

What happens to the graph as the standard deviation increases? What happens to the graph as the standard deviation decreases?

As the standard deviation increases, the curve gets flatter As the standard deviation decreases, the peak value of f(x) increases aka the midpoint gets higher

Explain how to find the area to the left of x for a normally distributed random variable X, using Table V.

If a normal random variable X has a mean different from 0 or a standard deviation different from 1, we can transform X into a standard normal random variable Z whose mean is 0 and standard deviation is 1. Then we can use Table V to find the area to the left of a specified z-score, z, as shown in Figure 5, which is also the area to the left of the value of x in the distribution of X. The graph in Figure 5 is called the standard normal curve.

List the four steps for drawing a normal probability plot by hand.

Step 1: Arrange the data in ascending order Step 2: Compute fi= (i-0.375)/(n + 0.25) where i is the index (the position of the data value in the ordered list) and n is the number of observations. The expected proportion of observations less than or equal to the ith data value is fi. Step 3: Find the z score corresponding to fi from Table 5 Step 4: Plot the observed values on the horizontal axis and the corresponding expected z scores on the vertical axis

What is a normal score?

The expected z score of the data value, assuming that the distribution of the random variable is normal. The expected z score of an observed value depends on the number of observations in the data set.

What happens to the graph of the normal curve as the standard deviation​ decreases?

The graph of the normal curve compresses and becomes steeper

What does the area under the graph of a probability density function over an interval represent?

The probability of observing a value of the random variable in that interval.

A​ _______ is a graph that plots observed data versus normal scores.

normal probability plot

What does the notation za represent?

(pronounced z sub alpha) is the z score such that the area under the standard normal curve to the right of zα is a.

Suppose that a random variable X is normally distributed with mean μ and standard deviation Give two representations for the area under the normal curve for any interval of values of the random variable X.

1. The proportion of the population with the characteristic described by the interval of values 2. The probability that a randomly selected individual from the population will have the characteristic described by the interval of values.

Explain how to determine if a normal probability plot is "linear enough".

Basically, if the linearly correlation coefficient between the observed values and expected z-scores is greater than the critical value found in Table VI, then it is reasonable to conclude that the data could come from a population that is normally distributed.

Things are not as easy for continuous random variables.

Because an infinite number of outcomes are possible for continuous random variables, the probability of observing one particular value is zero.

What happens to the graph as the mean increases? What happens to the graph as the mean decreases?

-As the mean increases, the graph of the normal curve slides right and x increases -As the mean decreases, the graph of the normal curve slides left and x decreases

For any continuous random variable, what is the probability of observing a specific value of the random variable?

0

State the seven properties of the normal density curve.

1. The normal curve is symmetric about its mean. 2. Because mean = median = mode, the normal curve has a single peak and the highest point occurs at x = μ. 3. The normal curve has inflection points at μ - σ and μ+σ 4. The area under the normal curve is 1 5. The area under the normal curve to the right of μ equals the area under the normal curve to the left of μ which equals 1/2 6. As x increases without bound (gets larger and larger), the graph approaches, but never reaches, the horizontal axis. As x decreases without bound (gets more and more negative), the graph approaches, but never reaches, the horizontal axis. 7. The Empirical Rule: Approximately 68% of the area under the normal curve is between x=μ−σ and x=μ+σ, Approximately 95% of the area is between x=μ−2σ and x=μ+2σ, and Approximately 99.7% of the area is between x=μ−3σ and x=μ+3σ.

A probability density function (pdf) is an equation used to compute probabilities of continuous random variables. It must satisfy the following two properties:

1. The total area under the graph of the equation over all possible values of the random variable must equal 1 2. The height of the graph of the equation must be greater than or equal to 0 for all possible values of the random variable. That is, the graph of the equation must lie on or above the horizontal axis for all possible values of the random variable.

Not all continuous random variables follow a uniform distribution.

For example, continuous random variables such as IQ scores and birth weights of babies have distributions that are symmetric and bell-shaped.

We discuss a _________ distribution to see the relation between area and probability.

uniform