QUANT: CHAPTER 9
The use of the sampling distribution underlies several ________ statistical techniques
Inferential
The mean of a sample is an ________ estimate of the population mean
Unbiased
When p is greater than (>) .05 it is NOT that unusual for our sample mean X bar to have come from a population with
mean mu= mu0
EX of how to write a conclusion statement for a confidence interval (CI)
"We can be 95% confident that the mean number of trials to criterion for neonates to learn to turn their heads in response to a tone is at least 43.51 and at most 49.15 trials"
Conclusion statements have 2 goals:
(1) write the conclusion in words that tell us the conclusion-what the problem or investigation is all about, including what is being measured (2) to convey the results in a way that someone unfamiliar with our problem- the "person in the street" can read the conclusion and have a reasonably good understanding of our results
Confidence intervals are useful in establishing:
-a range of scores within which we can safely expect a population value to occur -also useful in making decisions about hypotheses
The confidence interval is:
-an inference -an interval estimate of the population mean
The results of a sampling distribution of means
-approximately normal in shape -have its mean as the mean of the population from which the samples were drawn
Larger sample:
-gives us more precision -result in smaller values of s x bar (error) and smaller t scores from table B (both of which result in a narrower range of values for the CI)
A frequency polygon is drawn from the frequency distribution, two properties are shown:
-mu little x bar = mu and that the distribution is symmetrical
The shape of the polygon is
-symmetrical -is approximately normal with large samples
The alpha level will always be set to
.05 or .01 (usually)
Properties of the sampling distribution of means
1) the mean of the all the sample means (mu sub x bar) equals the mean of the parent population (mu). 2) the larger the size of each sample selected from the population, the more nearly the sampling distribution of means will approximate the normal curve (simplified version of the central limit theorem) 3) the larger the size of each sample selected from the population, the smaller the standard deviation of the sampling distribution of the means
2 distributions for confidence intervals & hypothesis testing
1.) the sampling distribution of means 2.) the t distribution * can be used to solve variants of the same two types of problems encountered in ch8 (finding areas when we know scores and finding scores when we know areas)
95 % confidence interval for the population mean
95% CI= +/- t.05 S x bar + x bar
99 % confidence interval for the population mean
99% CI= +/- t.01 S x bar + x bar
Statisticians usually favor what CI?
99% or 95% certainty
An alternative to the null hypothesis is that the sample came from...
A different population, which has been affected by some treatment or different condition and has some other mean that is not equal to mu o, called the alternative hypothesis
We do not have one t distribution as we do with the z distributions, but...
A family of t distributions *one for every degree of freedom (N -1)
Sampling distribution of means
A frequency distribution showing all possible sample means that occur when samples of a particular size are drawn from a population
The range of values is called a
Confidence interval
Margin of error is one common use of
Confidence intervals
In our conclusion, p tells us
About degree of certainty
Estimated standard error of the mean
An estimate of the true standard error obtained by dividing the sample standard deviation by the square root of the sample size
The rejection rule is...
An explicit rule rule for making a decision about our null hypothesis (whether to accept it or reject it)
The conclusion is
An inference statement about population parameters
Any one sample mean is more likely to differ from the popualtion mean than...
Be = to it
The larger the size of each sample we take from the population, the closer the sample mean will....
Be to the mean of the population
Non-directional hypothesis is sometimes called the
Bidirectional hypothesis (it takes into account both tails of the distribution) becuase the direction of the difference is not specified
When we talk about the deviant 5% or 1% of the distribution, we are considering....
Both ends of the distribution, or both tails of the distribution of t scores
The sampling distribution of means and the t distribution can be used to construct...
Confidence intervals (intervals of values around the sample mean) and to test hypotheses about the population from which our sample was drawn
Conclusion statements can be used for
Confidence intervals and hypothesis-testing problems
Hypotheses CAN NOT be confirmed TRUE, but can be...
Confirmed false
A 2 tail test
Considers both ends of the tails of the distribution of the test statistic (each of the values of t in the table is assumed to be both positive and negative)
A 1 tail test
Considers only one end of the distribution
Sometimes the sample mean will be higher than the population mean and sometimes it will be lower, but it will NEVER be
Consistently high or low *this is the property for which the sample is is an unbiased estimate of the popualtion mean
The t values table is a summary of the critical values of t that we will need for many __________
DF
We must know ______ to look up the critical values of t
DF
The disc tribute on is a family of curves, one for each...
Degree of freedom
With large samples, there is less _______ of the sample means from the population mean-that is, less variability about the population mean
Deviation
The alternative hypothesis can be
Directional or non directional
The sampling distribution of means is constructed by...
Drawing a sample of a certain size ( EX, N=30), calculating the mean of the sample, repeating the process for a large number of samples, then plotting the resulting means in the form of a frequency polygon
Estimates contain
Error
The t distribution is the normal distribution with ______ added
Error
When we estimate the standard error ( s little x bar), we are employing a particular sample size (N) to obtain that...
Estimate
In the social sciences, the problem most often faced by investigators is one of...
Estimating a population value from a sample value
Confidence Interval (CI)
Given a sample from a population, the CI indicates a range in which the population mean is believed to be found. Usually expressed as a 95% CI, indicating the lower and upper boundaries
The null hypothesis is written in general terms as:
H0: u = uo
We will compute a t value to tell us
How deviant our sample mean is
The alpha level we set defines...
How deviant the score must be in order for us to reject H0
Unbiased
If we draw a sample from a population and calculate the sample mean ( x bar), it will, on average, be the same as the population mean (mu)
The dispersions of the sampling distributions decrease as the sample size (N) used for each ______
Increases
Z score for a sample mean
Indicates how far sample mean is from population mean
The confidence level speaks to the degree of certainty we have in an
Interval estimate
We use t scores in the same way we use z scores, t x bar tells us how far a score (X) is from the _____ (mu) in estimated standard error units
Mean
They vast majority of sample means differ from each other. It is unlikely that any two successive samples will have precisely equivalent __________
Means
Non-directional hypothesis
Mu (pop mean) is not = to u o, but it does not specify in which direction it differes
When p is less than (<) .05 it exceeds the cutoff set by our alpha level, and we decide that IS very unusual for x bar to have been sampled from a population with...
Mu=mu0
The _____ chosen for the formula is the basis for that particular sampling distribution of means, which means that every sample size (N), we obtain a slightly different t distribution.
N
The standard deviation (standard error) decreases with increases in
N
The distribution of t more and more closely approximates the normal curve as
N increases
As the sample size increases, the sampling distributions become more....
Narrow and more sharply peaked, as well as more like a normal curve
Because the amount of error contained in an estimate becomes smaller as the sample size becomes larger, the t distribution becomes more and more similar to the _________ as sample size increases
Normal distribution
Our H1 is a ______ statement
Not equal
When we collect our data, we rarely have the luxury of knowing population __________
Parameters
What is the difference between calculating the z score from a population vs a sample?
Popualtion deals with sample means and popualtion deals with raw scores
Parent population
Population from which a sample has been obtained
The larger the sample we use in estimating the popualtion value, the closer our sample mean is likely to be to the....
Population mean EX: coin tosses
It is very unlikely that any given sample mean will exactly equal
Population mean (mu)
We can use our sample mean ( x bar) to estimate ____________, because x bar is an unbiased estimate of mu
Population mean (mu)
S (the sample standard deviation) is also an unbiased estimate of ______________. We can use s to obtain an estimate of the standard error of the mean ox, simply by substituting s for o in the equation ox
Population standard deviation ( o)
RARELY, will estimates exactly match the...
Population value
The confidence interval statement is not a
Probability statement
Hypotheses can never be ____________, only supported
Proved
The alternative hypothesis is also called
Research hypothesis and experimental hypothesis
Successive random samples of a given size are drawn from a (parent) population and a mean is computed for each ___________
Sample
A frequency distribution is constructed from the
Sample means
T more closely approximates z as ______ increases
Sample size
The frequency polygon is called the
Sampling distribution of means (also written sampling distribution of big X bar)
The alternative hypothesis
Says "No" to the null hypothesis
The sampling distribution will approximate the normal curve with larger sample sizes, regardless of the....
Shape of the parent population
When Ho is rejected, statisticians say that a __________ result has been found, corresponding to either the 5% or the 1% level of significance depending on the level chosen
Significant
The size of the confidence interval is larger with
Smaller samples
Estimates may miss the true (population) value by...
Some amount
Because the errors are the basis of a "standard" deviation (of the sampling distribution), we call it:
Standard error
The smaller variability with larger samples is reflect6ed in a smaller _________________
Standard error
The standard deviation of the sampling distribution of means is called the:
Standard error of the mean
Directional hypothesis
States the direction of the difference between mu and the sample mean, that is H1, may say that u0 is greater than u or less than u.
The T distribution is also called the
Student's t distribution because of William Sealy Gosset who published under "student" -Guinness beer and the barely variable -gosset needed to know how to make the barely less variable so he created the t distribution for use with small samples and unknown population variability -went under student so Guinness wouldn't be known for having a bad batch occasionally
A ____ score is an estimate and a t distribution incorporates the variability, or error that these estimates contain
T
We always take the absolute value of
T comp
In order to conduct a directional test:
T comp should have the same sign as t crit
T distribution
T scores
Where do we get t scores?
Table B
T can be used to test
The hypotheses about the value of mu (pop mean)
What happens to the confidence interval when the level of confidence level goes up?
The larger the CI, (say from 95% to 99%), there is a greater range of values, and the interval increases in size
Uo is
The mean of the population from which we assume our sample came
Degrees of freedom (DF)
The number of values in a set of scores that are free to vary after certain restrictions are placed on the data (For ex: if we are given 5 scores, only 4 of the scores are "free to vary", because the given sum determines the value of the 5th score)
The mean of the sampling distribution means (mu little x bar) is equal to:
The parent population (mu) and the standard deviation (called the standard error) becomes smaller as the sample size increases
We are more likely with large samples to obtain a distribution of scores that accurately reflects...
The population situation
Sometimes estimates are high or low but RARELY are they identical to
The population value
A CI is....
The range of values whiting which we are reasonably certain the population mean lies
For a t test, it approaches the normal curve as
The sample size increases
Central Limit Theorem (CLT)
The sampling distribution derived from a simple random sample will be approximately normally distributed
Most of the statistics we are given are only estimates because...
They are based only on some portion (a sample) of the larger group of intrest (population)
The null hypothesis or Ho states that
We assume the sample came from a population with a mean = to some some value ( mu o), which represents the usual state of affairs
The use of z scores for sample means is appropriate when
We know the population mean and population variance *our problem is that we RARELY know both of these values, so we must estimate the population values from our sample
The alternative hypothesis is performed to test to see...
Whether our hunch is correct about our hypothesis
The null hypothesis has to be put into
Words into the context of the problem
A ___ score is an exact value and the the normal curve is a distribution of these exact units
Z
The sum of the deviations of scores is always equal to
ZERO
Null hypothesis
a statement or idea that can be falsified, or proved wrong
Directional vs non-directional hypothesis
directional - has a clear bias non-directional - no clear bias
The rejection rule
is that H0 is rejected in favor of H1 at the 5% significance level if t>c
DF=
n-1
The alpha level
the probability level used by researchers to indicate the cutoff probability level (highest value) that allows them to reject the null hypothesis
A score for a sample mean formula.
z = (x - μ) / σx (standard error)
The standard error of the mean (ox) is = to
σ/ square root of N *we know this because of the third property of the sampling distribution (tells us that the larger the size of each sample selected from the population, the smaller the SD of the sampling distribution of the means
