Lecture 6_9

standard, mean, standard deviation

Any normal distribution can be transformed into the __________ normal distribution with 2 steps: 1. subtract _______ 2. divide by _________ _________

increases

As df __________ (increases or decreases), t distribution becomes the Z distribution.

t distribution, z distribution

As the sample size increases, the ___ __________ becomes the ___ __________.

mean, variance

Calculating areas allows us to predict how much of the population data is in a region if we have a sample ______ and _________.

t table, sample, sample s

Considering a more realistic situation in which the population standard deviation is unknown and we have to estimate is from the sample data, 1. Which table will be used? __ _______. 2. What are the first two calculations made? ________ mean and __________ __

right

For our t table, the α refers tot he area to the _______ (right or left). * Note: Our z table shows area to the left.

α/2

How do you find the tcrit value? t ____, df

Z

If we know σ, the __ distribution can be used.

means

If we study _______ we can use the normal distribution.

No

If µ known and σ is known, is sampling needed? Answer yes or no.

z distribution

If µ unknown and σ is known,what kind of distribution is used? ___ _________.

t distribution

If µ unknown and σ is unknown,what kind of distribution is used? ___ _________.

Z

If σ is known and µ is unknown, we use values from a ___- Distribution since σ is known.

µ, T

If σ is unknown and ___ is unknown, we use values from a ___- Distribution since σ is unknown. * Note the "t distribution is wider than the Z to include our uncertainty in estimating σ with s.

T

If σ is unknown, then the ___ distribution must be used.

normal distribution

Many populations exhibit a ________ __________.

standard error

Measures spread of potential of the population mean. What is this called? ___________ _______.

standard deviation

Measures spread of sample or population data values. What is this called? _________ ________.

standard error

S (xbar) = σ÷√n This is the __________ _______ of the sample mean.

mean

Sample means ~N(_______, variance^2/ n) * Note: The variance of the distribution of sample means depends on the size of the sample.

s (aka sample standard deviation)

Since we usually don't know "σ", we have to estimate it from "___". * This means we can't just use z-scores, we have to use t distribution, which includes uncertainty in our estimation of σ via s.

alpha

Table of z-scores: Values in the table are α (_________) values.

z score

The __- ________ allows comparison or standardization of data sets with different means and standard deviations.

Normal Distribution

The area under the curve between two x values represent how much of the data is in that range. What kind of distribution is this? __________ ___________.

confidence interval

The range around the sample mean is a ___________ _________. We are x% confident that the true population mean lies within a region around our sample mean that includes x% of the distribution.

confidence intervals

The ranges around the mean are ____________ __________. We are confident the true population mean lies within a region around our sample mean. Bigger region means more confidence but less utility.

uncertainty

The t distribution includes the __________ in estimate of σ and is wider than the z distribution.

size, variance

The variance of the distribution of sample means depends on the ______ of the sample and population _________.

True

True or False: If the degrees of freedom are not listed on the t table, always round down to the next closest number.

True

True or False: The goal is the describe a region within which we are confident the population mean lies.

False

True or False: The goal is the describe a region within which we are confident the sample mean lies.

True

True or False: To find the t critical value, you need alpha and the df (i.e. t α,df)

True

True or False: Tables often require interpolation if sample size is not listed.

quantitative, sample

Using the Normal distribution allows calculation of confidence intervals and ___________ statements about population mean from ________ data.

n

Using the t distribution also requires us to specify the degrees of freedom (df) of the data, in this case, df= __ -1.

mean, deviation

We typically don't know population _______or population standard ___________( if we did we wouldn't need to get a confidence interval because we would know µ) so we use "X bar" to estimate µ and "s" to estimate σ.

confidence interval

What is this formula for? ___________ ___________ Xbar +,- tcrit (σ/√n)

confidence interval

What is this formula for? ___________ ___________ Xbar +,- zscore (σ/√n)

larger, narrower

When using a z distribution to find a confidence interval where the true population mean lies, a ________ sample size means that the CI for the same degree of confidence is __________ because the SE is smaller.

normal

Why do we care about areas under the Normal Distribution? Because of the Central Limit Theorem (CLT). For large sample size the distribution of sample means will be _________, no matter what the actual population distribution is.

right

Z 1-α is the value Z such that there is α area to the _________ (right or left).

right

Z 1-α/2 is the value Z such that there is α/2 area to the _______ (right or left).

left

Zα is the values Z such that there is α area to the _______ (right or left).