Stat 250 Exam 2

statistical inference

the process of drawing conclusions about the entire population based on the information in a sample

parameter

used to identify a quantity measured for the population

a number that describes some aspect of a population.

statistic

used for a quantity measured for a sample

a number that is computed from the data in a sample.

notation for mean (population parameter)

mu

notation for mean (sample statistic)

x-bar

notation for proportion (population parameter)

p

notation for proportion (sample statistic)

p-hat

notation for standard deviation (population parameter)

sigma

notation for standard deviation (sample statistic)

s

notation for correlation (population parameter)

"rho"

notation for correlation (sample statistic)

r

notation for slope (population parameter)

Beta

notation for slope (sample statistic)

b

point estimate

We use the statistic from a sample as a point estimate for a population parameter.

If we only have the one sample and don't know the value of the population parameter, this point estimate is our best estimate of the true value of the population parameter.

sampling distribution

the distribution of sample statistics computed for different samples of the same size from the same population.

A sampling distribution shows us how the sample statistic varies from sample to sample.

shape and center of a sampling distribution

For most of the parameters we consider, if samples are randomly selected and the sample size is large enough, the sampling distribution will be symmetric and bell-shaped and centered at the value of the population parameter.

standard deviation

is a statistic that measures how much variability there is in the data- measures the spread

measures theaverage/typical distance from all the observed data to the mean

is the standard deviation of the individual values in that one sample

standard error

denoted SE

is the standard deviation of the sample statistic-measures variability

a "typical" distance between the sample statistics and the population parameter

the standard deviation of the sample statistics if we could take many samples of the same size

95% Rule

If a distribution of data is approximately symmetric and bell-shaped, about 95% of the data should fall within two standard deviations of the mean. This means that about 95% of the data in a sample from a bell-shaped distribution should fall in the interval from

The 95% rule can be used for distributions that are..

symmetric and bell shaped

For the 95% rule, 95% of the data in a sample should fall in the interval from...

xbar - 2s to xbar + 2s

or mu-2sigma to mu+2sigma

A low standard error means..

statistics vary little from sample to sample, so we can be more certain that our sample statistic is a reasonable point estimate

What does a larger sample size mean?

As the sample size increases, the variability of sample statistics tends to decrease and the smaller the standard error of the sample statistic

sample statistics tend to be closer to the true value of the population parameter.

Statistical inference caution

Statistical inference is built on the assumption that samples are drawn randomly from a population. Collecting the sample data in a way that biases the results can lead to false conclusions about the population.

interval estimate

gives a range of plausible values for a population parameter

equation for interval estimate

Point estimate ± margin of error

margin of error

a number that reflects the precision of the sample statistic as a point estimate for this parameter.

is the amount added and subtracted in a confidence interval

confidence interval

for a parameter is an interval computed from sample data by a method that will capture the parameter for a specified proportion of all samples.

confidence level

The success rate (proportion of all samples whose intervals contain the parameter)

If we can estimate the standard error SE and if the sampling distribution is relatively symmetric and bell-shaped, a 95% confidence interval can be estimated using

Statistic ± 2(SE)

The STD, SE, and margin of error are all..

DIFFERENT

The confidence level indicates how sure we are that..

our interval contains the population parameter. For example, we interpret a 95% confidence interval by saying we are 95% sure that the interval contains the population parameter.

We are 95% confident that the... .. is in the interval.

population mean

we are 95% sure that the mean of the...will fall within a 95% confidence interval for the mean

population

A confidence interval gives us information about the..

population parameter

bootstrapping

allows us to approximate a sampling distribution and estimate a standard error using just the information in that one sample

bootstrap samples

sampling with replacement from the original sample, using the same sample size.

bootstrap statistic

the statistic of interest for each of the bootstrap samples

bootstrap distribution

create a distribution by collecting the statistics for many bootstrap samples

To generate a bootstrap distribution we..

Generate bootstrap samples by sampling with replacement from the original sample, using the same sample size.

Compute the statistic of interest, called a bootstrap statistic, for each of the bootstrap samples.

Collect the statistics for many bootstrap samples to create a bootstrap distribution.

The standard deviation of the bootstrap statistics in a bootstrap distribution ....

gives a good approximation of the standard error of the statistic

When a bootstrap distribution for a sample statistic is symmetric and bell-shaped, we estimate a 95% confidence interval using

statistic ± 2(SE) where SE denotes the standard error of the statistic estimated from the bootstrap distribution

If the bootstrap distribution is approximately symmetric and bell-shaped....

we construct a confidence interval by finding the percentiles in the bootstrap distribution so that the proportion of bootstrap statistics between the percentiles matches the desired confidence level.

A larger sample size tends to .....

increase the accuracy of the estimate, giving a smaller standard error and reducing the width of a confidence interval.

The bigger the sample..

the more narrow the confidence interval