Albert MorlanCaitrin Carroll

Savannah AndrewsRichard Saney

Variables

Here are some variables that you need to know:Sample Mean:Population Mean:Sample Proportion: Sample Standard Deviation:Population Standard Deviation: Sample Size: n

X

X

xSX

p̂

Review from Last Year A statistic comes from a sample. A

parameter comes from a population. Standard deviation is the average distance

from the mean. Quantitative data has a numerical value.

Qualitative data does not. Z-Scores are used to calculate area below

that point on a normal distribution curve. These show the amount of standard deviations

a statistic is from the mean.

Sampling Distribution

A distribution of all possible values of the mean from a given sample size

The mean of the sampling distribution is a good estimator of the mean of the population.

The standard deviation of the sampling distribution is :

Or for qualitative data:

X

xS

nˆ ˆ(1 )p p

n

T-Distribution• The T-Distribution is used when you don’t

have the standard deviation of the population. The standard deviation is replaces by the standard area of the mean of the sample. It is only used for quantitative data.

• The T-Distribution varies for each sample size, and the distribution is determined by the degrees of freedom. The more degrees of freedom, the more normal the distribution.

• df=n-1

T-Distribution (cont)

In a T-Distribution there is more area in the tails than in that of a Normal Distribution (Z-Distribution).

Central Limit Theorem (CLT) Not all sampling distributions are

normally distributed. The CLT states that the larger the

sample size gets, the more normal the sampling distribution becomes.

Estimating the Mean

To estimate the mean of the population from a single sample, perform a confidence interval.

To perform a confidence interval, P.A.N.I.C

P.A.N.I.C PANIC is a acronym that stands for:

ParameterAssumptionsName the intervalInterval (actually do it)Conclude in context

Parameters

Name the parameters of the distributionEx: is the true population mean grade

of students in Coach Whitt’s classEx: px

X

Assumptions

Several Conditions have to be met to run a confidence interval:The data has to have come from a simple

random sample○ This is usually given in the problem. If not,

you have to assume this is true.

Assumptions (con’t)The sampling distribution has to be

approximately normal○ If the data is quantitative, construct a normal

probablity plot ( a graph of data against it z-scores)If the sample size is large (greater than 30), normal it

is ensured by the CLT○ If the data is qualitative, there has to be at

least 10 “successes” and ten “failures”.Each data point has to be independent

○ If the population size is at least ten times greater than the sample size, independence is ensured.

Name the Interval

Decide which interval will be usedIf the data is quantitative, use a 1-sample t

intervalIf the data is qualitative, use a 1-proportion z

interval

Also, determine the desired confidence level.How confident do you want to be is your

estimation?

Interval The formula for the confidence interval is:

(test statistic) (margin of error) Test statistic is the value obtained by the

sample (either or )

X p̂

Interval (con’t)

(1-confidence level)

2

Margin of error is (critical value * standard deviation)

Critical ValueFor quantitative data, t*For qualitative data, z*

○ Both obtained either from the table or from the calculator (invNorm(area) or invT(area, degrees of freedom) Area =

Statdard deviation is that of sampling distribution

Conclude in Context

Be sure to write a conclusion is context of the problem

Ex: I am 95% confident that the true mean grade of students in Coach Whitt’s class is between 80 and 90.