Chapter 7 - Sampling Distributions

What is the "Nearly Normal" condition?

one of three things have to be true: - the population is known to be bell-shaped - the sample data are bell-shaped - the sample size is at least 30 (Central Limit Theorem)

The center of the sampling distribution for x-bar is the value of the...

parameter (in this case, mu, the population mean)

when σ is known, we calculate SE(x-bar) using ___________, and we use the ____________ to mode the sampling distribution for the sample mean.

population standard deviation; normal curve

If you are ever asked about the probability associated with a sampling proportion taking certain value, you are being asked about...

a sampling distribution

The sampling distribution is defined by the...

it is defined by the mean and the standard deviation

the normal distribution is symmetric, so it's center is the...

mean

we call the standard deviation of the sampling distribution the...

standard error

What does SE(p-hat) mean?

standard error of p-hat (we do not multiply the standard error by p-hat)

When we want to standardize a value, we use an equation of the form...

standard statistic = observation-mean of observation s distribution/SD of observation n s distribution

For large samples sizes, the t- and z-curves are very similar, but the _________ is noticeably wider for small samples sizes.

t-curve

When you use the standard deviation of the sample to estimate standard error, the subsequent sampling distribution for x-bar uses...

t-distribution rather than normal distribution

When using stat crunch to find sampling distribution, by clicking "1 time" once we are...

taking a single sample.

(sampling distribution of sample mean) Center?

the center of the sampling distribution is ALWAYS the value of the parameter - population mean.

(sampling distributions for proportions) Center?

the center of the sampling distribution is ALWAYS the value of the parameter - population proportion.

sampling distribution

the collection of all possible sample statistics we could attain from samples of the same size from a population.

success/failure condition

the sample size must be big enough so that both the number of "successes," np, and the number of "failures," np, are at least 15.

When σ is unknown, we calculate SE(x-bar) using __________. This adds a little bid more uncertainty to our estimate, so we model the sampling distribution for the sampling mean with the _________.

using 's' in it's place; t-distribution

How do you find the standard error of the mean?

we prefer to estimate SE(x-bar) using mu/square root of n, but oftentimes we do not know the populations SD, so then we would use s, the sample standard deviation.

The sampling distribution for x-bar will be __________, under certain conditions, like the sampling distribution for p-hat.

bell-shaped

Regardless of sample size, the __________ will be the value of the parameter.

center of the sampling distribution.

In particular, we will estimate the sampling distribution of the sample mean for the ______________.

centralized t-curve

When the standard error is small, we expect to get sample proportions ________ the population proportions.

close to

when the standard error is small, we expect to get a sample mean ______ the population mean.

close to

What are degrees of freedom?

conceptually, they are the number of data points you need to know until you can fill in the remainder f the dataset with certainty. for now, know that DF = (n)-1

As the degrees of freedom increase (i.e. the sample sizes get larger), the t-distribution...

converges upon the standard normal

As n increases, the standard deviation...

decreases

When we increase n (sample), the standard deviation of a sampling distribution ________.

decreases

as n increased, the standard deviation of the sampling distribution...

decreases

Our t-distributions are defined by...

degrees of freedom

When the standard error is large, we expect to get some _______ the population proportion.

far from

when the standard error is large, we expect to get a sample mean ______ population mean.

far from

As n decreases, the standard deviation...

increases

as n decreases, the standard deviation of the sampling distribution...

increases

we require that the samples be __________ and ___________.

independent and randomly selected

When we have categorical data, we can calculate...

a proportion. (the number of subjects in a specific category)

Sufficient Sample Size Assumption

- the sample size, n, must be large enough that the sampling distribution is not truncated at either end. (which is generally satisfied by the 15 successes)

Randomization and Independence Assumption

- the sampled values must have been randomly selected or result from experiment with random assignment - they should be independent of each other *generally any form of randomization will also ensure that the independence facet is satisfied.

(sampling distributions for proportions) When to use?

1) categorical being measured; 2) we are being asked about the probability associated with a certain sample proportion value

(sampling distribution of sample mean) When to use?

1) quantitative variable being measured; 2) we are being asked about the probability associated with a certain sample mean value.

Two conditions must be satisfied for the collection of p-hat's (the sampling distribution of p-hat) to be approximately normal:

Randomization and Independence Assumption, and Sufficient Sample Size Assumption

To calculate the standard error for the sampling distribution of x-bar, we find:

SE(x-bar) = standard deviation/square root on n

What is the center of a sampling distribution?

The center is the value of the parameter, so we know the mean of the sampling distribution of p.

(sampling distributions for proportions) Spread?

The standard deviation for the sampling distribution (standard error) for a proportion is calculated as SE = squareroot [p(1-p)] / n

In the z-score formula, we call our observation ______, the mean of the population from which we draw that observation is _____, and the standard deviation is called _____.

observation - x ; the mean of the population - mu ; SD - σ

(important property of sampling distribution) The center of any sampling distribution will always be the value of the ___________ , regardless of sample size.

population's parameter

's' and 'σ' mean...

s - standard deviation of the sample σ - standard deviation of the population

When we work with a sampling distribution for x-bar, our 'observation' is actually a ________.

sample mean

If a question is phrased "if I take a sample of this size, what is the PROBABILITY THAT THE SAMPLE PROPORTION will be greater than or less that the same value?" we know it is...

sampling distribution

When using stat crunch to find sampling distribution, the bottom graph that appears with the lines is forming a...

sampling distribution.

any individual statistic we obtain must be somewhere on our...

sampling distribution.

What is x-bar?

sampling mean

When we randomly select a sample from a population, we are, in effect,...

simultaneously selecting a statistic from the sampling distribution.

To identify the SD of the sampling distribution for p-hat, distributions that are bell-shaped and centered at the true proportion, p, we can use...

the sample size, n, to find the standard deviation of the sampling distribution (standard error): SE(p-hat) = square root [p(1-p)] / n (categorical variables)

(sampling distribution of sample mean) Shape?

the sampling distribution for the sample mean will be approximately normal if both of the following are true: - the sample was selected randomly and independently - "Nearly Normal" condition

(sampling distributions for proportions) Shape?

the sampling distribution for the sample proportion will be approximately normal if both of the following are true... - the sample was selected randomly and independently - successes and failures condition: np > 15

(sampling distribution of sample mean) Spread?

the standard deviation for the sample distribution (standard error) for a mean is calculated as SE = SD of population (or sample)/square root of n

the mean of a sampling distribution will always equal ______________ from which the samples are drawn.

the value of the parameter