Reading Quiz 12 Flashcards
steps for significance tests in general
- state the parameter and population, null and alternative hypothesis, and significance level
- state the inference procedure to be used (or state with calculations) and verify its conditions (SRS, normal, independence)
- complete all necessary calculations: z or t statistic formula, values in, z/t statistic value, and p value
- state conclusion/interpretation, make sure in context
normality for sample means
normal if population distribution normal
approx normal by central limit theorem if n greater than or equal to 30
or observe and sketch histogram
normality for sample proportions
npnot and nqnot greater than or equal to 10
independence additional evidence
10n less than or equal to N
significance tests for mean μ of a normal population
based on sample mean x̅ of an SRS of size n
thanks to central limit theorem, resulting procedures are approx correct for other population distributions when sample is large (greater than or equal to 30)
when we know σ
use z statistic and standard normal distribution to perform one-sample z test for means
when don’t know σ
use one-sample t statistic, which is used for one-sample t test for means
t statistic has
t distribution with n-1 degrees of freedom
as degrees of freedom increases, shape of t distribution
more and more closely approximates the standard normal distribution
t statistic interpretation
says how far away x̅ is from its mean μ in standard deviation units
significance tests for Ho: μ = μo are based on
t statistic
use p values or fixed significance levels from the t (n-1) distribution
paired t test
use to analyze paired data by first taking the difference within each pair to produce a single sample
then use one-sample t test procedures
aka one-sample t test on the differences
power of a t test
hard to calculate, rely on technology
measures ability to detect deviations from the all hypothesis
higher power important
z statistic for tests of Ho: p = pnot
z = (p̂ - pnot) / (sqrt (pnot * qnot)/n) )
one-sample z test for proportions
z statistic z = (p̂ - pnot) / (sqrt (pnot * qnot)/n) ) used with p values calculated from the standard Normal distribution
important note
use pnot in the denominator for the z test statistic and not p̂. probability calculations assume Ho is true, and since Ho: p = pnot, assuming that p = pnot
large sample test
often called one-proportion z test
called large sample because based on Normal approximation to the sampling distribution of p̂ that becomes more accurate as the sample size increases
what additional information do confidence intervals provide that significance tests do not
a range of plausible values for the true population parameter p
for inference about a population proportion, confidence intervals and two tailed tests
do not have the perfect correspondence that happens with inference about a population mean, but relationship is similar
The one sample z-statistic is the z = (xbar - mu not)/(sigma/sqrtn). What is the one sample t-statistic equation?
t = (xbar - mu not)/(s/sqrtn)
The one-sample t statistics has the t distribution with ________ degrees of freedom.
n - 1
When using the t distribution critical values chart to determine the p-value corresponding to a particular t statistic, if you have degrees of freedom = 58, since there isn’t a df = 58 row should you go down and use the df = 50 values or round up to df = 60?
Always round down if the exact df is not available, in this case you should use the df = 50 values.
When two sets of data is paired and you take the difference within each pair to produce a single sample, what are you performing?
a paired t test
Suppose someone were to draw many pairs of samples from two populations, and compute the difference between the sample means for each pair. What would the mean of this difference approach as the number of samples drawn approached infinity?
the difference in population means
The fact that the mean of the difference in sample means approaches the difference in population means as the number of samples gets larger is a long way of saying that the difference in sample means is an ____ estimator of the difference in population means.
unbiased
Something to remember from chapter 10: what is a “robust” procedure? Hint: page 654.
One where the accuracy of a confidence interval or significance test is not very greatly affected by violation of the assumptions.
The t procedures are very robust against (non-normality of the population, outliers) but not very robust against (non-normality of the population, outliers).
Non-normality of the population, outliers
The main reason why the t procedures are robust against the non-normality of the population is what theorem?
The central limit theorem, i.e. that sample means become more nearly normally distributed as the sample size gets larger, even when the population does not have a normal distribution.
The statistic that estimates (in an unbiased way) the population proportion is ____.
the sample proportion
What is the equation for a one proportion z statistic?
z = (p̂ - pnot) / (sqrt (pnot * qnot)/n) )
If npo and nqo are at least 10, then we can treat the distribution of p-hat as _________________.
approximately Normal. In other words, the Normality condition is then considered met. It is
important to notice that it npo not np-hat.
Instead of the method described in the previous problem, can the central limit theorem be used to verify the Normality condition for one-proportion z test?
No, the central limit theorem can only be used with means, this deals with proportions!
What additional information do confidence intervals provide that significance tests do not?
Confidence intervals provide a range of plausible values for the true population parameter p.
