ESTIMATING THE POPULATION MEAN WHEN THE POPULATION STANDARD DEVIATION IS UNKNOWN Flashcards
Because we don’t know what the population standard deviation is, what do we use instead of a z distribution to estimate the population mean?
A t-distribution
When the population standard deviation is unknown, what point estimate to we use for it?
The sample standard deviation
What is another word for t distribution? When do we rely on it?
Student’s t-distribution (we assume for now that the population is normally distributed)
when sample size is small (less than 30) and/or when the population standard deviation is unknown.
Why are extreme scores (lots of tres bas or tres high scores) more likely to occur in a t distribution than in a normal distribution?
Because when estimating the population standard deviation, we lose some information; less true information and more room for error.
Also more extreme scores means less height compared to standard normal distribution (diapo 6)
What are two characteristics of the t distribution?
- Similar to normal distribution, but with thicker tails; this gives rise to more extreme cases
- There is a different t distribution for each number of degrees of freedom
a) the more degrees of freedom on which the t distribution is based, the closer it is to a normal curve
(WHY? more information about the population)
b) when there are infinite degrees of freedom the t distribution is the same as the normal curve
* (WHY? because your sample is infinitely large, and thus is the same as the population)
- For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value.
- The standard normal z values can be found in the infinite degrees (∞ ) row of the t distribution table.
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EXAM: You lose […] degree of freedom for every sample mean you calculate.
analogie: 5 bags, mean 10lbs, all 4 bags free to vary for the last one HAS to be a certain number (fixeD) for the sample mean (10lbs) to be true.
How do you determine the cutoff scores on a t-distribution? What are three things you need?
You use the t-table.
To use the table, you need degrees of freedom (n-1), significance level (alpha or confidence level), and whether you are using 1 or 2 tail (only applies when doing hyp testing with alpha level) .
Since a sample represents its population, a sample’s standard deviation represents/is representative of…
The population standard deviation
Can we use a sample’s standard deviation (s) as a direct estimate of the population variance (o2) ? 2 reasons.
NO. use s2 instead as s is a biased estimator.
- The sample standard deviation (s) is calculated from a subset of the population.
On average, it ends up being slightly smaller than the true standard deviation of the population (σ).
This makes it a biased estimate, meaning it doesn’t perfectly represent the population standard deviation - Sample standard deviation is a biased estimate of the population standard deviation
- What we need is, you guessed it, an unbiased estimate of the standard deviation ( “unbiased” means the estimate, on average, matches the true value of the population parameter)
- Sample variance (s2) is an unbiased estimate of population variance (σ2)
However, sample standard deviation (s) is biased for estimating the population standard deviation (σ). Variance (s2) is an unbiased estimate because of the use of
n−1, but the square root transformation to get the standard deviation (s) reintroduces bias.) - Why Do We Still Use
s?
Despite the bias, we often use the sample standard deviation because it’s useful and interpretable. Adjustments (like dividing by
n−1 instead of n) help make variance calculations unbiased.
degrees of freedom means what?
number of scores that are free to vary when calculating the standard deviation (n-1)
How do we calculate sample standard deviation?
Square root of (SS/df)
Once you know s, you can figure out the standard deviation of the comparison distribution which is…
Sm = s/ racine de n
(Au lieu de o/racine n)
what is : T critical x Sm ?
The sampling error.
With an unknown pop standard dev, how do we calculate the confidence interval estimate of pop. mean?
CI = Sample mean +/-
(tcritical x Sm)
If we have a confidence level of 90%, and we found a sampling error of 0.225. How can we phrase the conclusion/interpretation?
For 90% of such intervals, the sample mean would not differ from
the actual population mean by more than 0.225 (this is your sampling error). WARNING: Never say
you are 90% confident the interval contains your population mean. The probability of this is either 0 or 1. It is either in there or it is not.
What are the regular steps in hyp testing. Which two steps are gonna differ for the t-test?
1- restate the question as a research hypothesis and a null hypothesis about the population
2- determine the characteristics of the comparison distribution
these 2 steps will differ for the t test
3- determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected
4- determine your sample’s score on the comparison distribution
5- decide whether to accept or reject the null hypothesis
STEPS 2 and 3 will differ.
What is the basic principle of the t test?
if the population standard deviation variance is not known a solution is to estimate it from the sample (biased estimator tho right??).
What’s a sample t-test?
- Same as when we were converting our sample scores to Z scores
- However the resultant score is called a t score instead of a Z score
formula for determining the t score of the sample mean:
t= (Sample mean - pop mean) / Sm (standard deviation of the t distribution (sampling distribution which follows a t distribution)
If p<alpha...>alpha...</alpha...>
Significant
Not significant
When you find the obtained t score, what are you trying to find?
The probability of getting that t score under the null. Then figure out if p< or > alpha.
How do you state your conclusion?
A t–test revealed that … differed (OR NOT) significantly from [POP MEAN], t(df) = tscore, p < or > 0.05, therefore we can reject/retain the null hypothesis
What does statistically significant mean?
The difference we’re seeing is less likely to be due to random chance.
f my statistics problem asks for t critical or z critical for two tails, do I give +/- value or just the absolute value??
You only need to report the positive critical value (absolute value), because the negative critical value will be the same in magnitude but opposite in direction due to symmetry.
For z-scores: You’ll only report the positive critical z-score.
For t-scores: You’ll also report only the positive critical t-score.
If the confidence interval does not include the mean of the null hypothesis (uo), we….
declare the results of the study to be significant.
(this is the approach which gives us a possible range of means under the alternative distribution!)
