Module 3 Flashcards
statistical inference
idea of using sample data to provide evidence about population parameters
what can we use to estimate unknown population parameters?
- Point estimates
- confident intervals
point estimate
single value used as an estimate of a population parameters
Xbar
sample mean
u
population mean
________ the sample mean is a point estimate of _________
xbar sample mean is a point estimate of u the population mean
s^2
sample variance
o^2
population variance
_______ the sample variance is an estimate of ______________
s^2 the sample variance is a point estimate of the o^2 the population variance
s
sample standard deviation
____ the sample sd is a point estimate of ______________________
s the sample sd is a point estimate of o the population sd
what are point estimates influenced by ?
sampling error
the variation of a point estimate from one sample to another represents the extent of ___________________
sampling error
Sampling error in relation to sample size
Smaller samples have more sampling error than larger samples so we expect point estimates from small samples to have more sampling error than point estimates from larger samples
formula for the standard error of the mean
represents the amount that the sample means are spread around the population mean of a variable - or amount that a sample mean is expected to differ from the population mean on average
oxbar = o/sqrtN
the bigger N gets the smaller oXbar gets - mean has less sampling error with larger N
confidence interval
range of values or an interval that is expected to capture a population parameter of interest with some prespecified level of confidence
presents a range of plausible or reasonable values for the population parameter
Confidence limits
specify the particular range of values for a confidence interval
smallest value = lower limit
biggest value = upper limit
confidence intervals in relation to sample size
confidence intervals from small samples are wider than confidence intervals from larger samples
central limit theorem tells us that the sample means can be treated as observations from a _____________
normal distribution
95% of all sample means fall within _______ standard error units of the population mean
1.96 standard error units of the population mean
the 95% confidence interval is expressed as what formula
Xbar +/- 1.96*oxbar
where 1-alpha is the level of confidence
oxbar is also (o/sqrtN)
Zalpha is what
critical valye
Critical value for a 95% confidence interval
1.96
Assumptions for a confidence interval calculated using the standard normal distribution
- Independent observations
- The sample data come from a normal population
- The population standard deviation is known
Assumptions for CI
- Independent observations
- Having info about one participant provides no further info about another participant
Assumptions for a CI
- The sample data come form a normal population distribution
Because of the central limit theorem, this assumption can be ignored as long as the sample size is large. How large depends on the extend of non-normality of the population distribution of individuals.
the width of a confidence interval represents _________
the precision of a point estimate with respect to its estimation of the corresponding parameter
Does the population mean change?
The population mean is a constant, fixed, but unknown value that does not change.
Does the confidence interval change?
The lower and upper confidence limits are random variables meaning they do change from sample to sample
If you have calculated a 95% confidence interval how do you phrase it
if the exact sampling procedure were repeated an infinite number of times and a 95% confidence interval were calculated around each sample, 95% of those intervals would contain u. The interval of blank to blank is one of these infinitely many intervals.
Easier:
The interval captures u with 95% confidence
The interval blank to blank captures the population mean number of drinks per week with 95% confidence
Factors affecting the width of a confidence interval that are under the researcher’s direct control
- Level of confidence - larger values of 1- alpha have wider intervals
- Sample Size - larger samples lead to narrower confidence intervals
Null hypothesis
represents another method for researchers to describe sample evidence about the true value of a population mean
“We reject the null hypothesis that the population mean equals 0 with probability = alpha that we are incorrect
want to know whether a mean is different enough to conclude what
conclude that the population mean of home schooled 4th graders is likely to be different from 250
if the population mean is u=250 how unlikely is a sample mean of xbar=272 or more extreme
Formal steps of a null hypothesis testing procedure
- Specify the null hypothesis
- Specify an alpha level
- 0.05 - Calculate a test statistic
- convert sample mean to z score xbar-uo/o/sqrtN - Calculate a P-value
- probability of obtaining a score greater or equal to the one we are comparing to - Compare the p-value alpha/2 to reach a decision about the null hypothesis
- two tailed test = have to divide alpha by 2
alternate hypothesis example
the population mean achievement score of home-schooled 4th graders does not equal 250
Non-directional which implies a two-tailed test
alpha level
probability of rejecting a true null hypothesis
test statistic
value calculated from a sample descriptive statistic that is used to evaluate the descriptive statistic relative to the population value specified by the null hypothesis
Does the p-value demonstrate the probability that the null is true?
NO
Correct interpretation of the p-value
Probability of obtaining the current sample mean or one that is more extreme under the assumption that the null is true.
Assumptions of one sample z-test
- independent observations
- sample data come from a normal population distribution
- the population SD is known
Effect size
difference between the sample mean compared to the hypothetical population mean
idea of by HOW MUCH a null hypothesis is likely to be wrong
Does statistical significance convey practical importance?
no, effect size does
