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AP Statistics Chapter 8 - Estimating with Confidence
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Terms in this set (13)
Confidence level C
The success rate of the method that produces the confidence interval. In C% of all possible
samples, the method would yield an interval that captures the true parameter value.
"If we were to take many, many samples of the same size from this population (context), about C% of them will result in an interval that captures the _____________(actual parameter value in context)."
Confidence interval
An interval calculated from sample data, which has the general form:
point estimate ± margin of error or, alternatively, statistic ± (critical value)(standard deviation of statistic).
The critical value depends on both the confidence level C and the sampling distribution of the statistic (z or t).
"We are C% confident that the interval from ________ to ______ captures the true p or mu of ______________________(in context)."
Margin of error
Tells how close the estimate tends to be to the unknown parameter in repeated random sampling. This is the (critical value)(standard deviation of the statistic).
One-sample t interval for a population mean
C% confidence interval is (x-bar) ± (t*)(Sx/sqrt(n))
where t* is the critical value for the t distribution with df = n - 1.
Conditions:
1) Random
2) 10% condition (shows independence)
3) Sampling distribution approximately Normal? the population distribution is Normal, the sample size is large (n ≥ 30) CLT or if those items aren't known, graph the sample data and look for clear outliers or strong skewness. If no clear outliers or strong skewness it is reasonable to assume Normality of population and continue to create the confidence interval.
One-sample z interval for a population mean
This is only used if we KNOW sigma (the population standard deviation).
This is not common, as we rarely know sigma for the population. Use z
value, rather than t
.
Point estimate
The specific value of a point estimator.
Point estimator
A statistic that provides an estimate of a population parameter.
Standard error
When the standard deviation of a statistic is estimated from data, the result is the standard error of the statistic.
Standard error of the sample proportion p^
([p^](1-p^)/n). It describes how far p^ will be from p, on average, in repeated SRSs of size n.
Standard error of the sample mean x
(Sx)(sqrt(n)) where Sx is the sample standard deviation. It describes how far x bar will be from µ , on average, in repeated SRSs of size n.
The t distribution; degrees of freedom
Standardized, centered at 0, more area in the tails, symmetric. Each t distribution varies based on the sample size.
Degrees of freedom is n-1. The larger the sample size, the closer the t distribution gets to the Normal distribution.
One-sample z interval for a population proportion.
C% confidence interval is
(p-hat) ± (z*)(sqrt((p-hat(1-p-hat))/n))
where t* is the critical value for the t distribution with df = n - 1.
Conditions:
1) Random
2) 10% condition (shows independence)
3) Large counts
np>= 10 and n(1-p)>=10. (shows Normality)
Construct and interpret a confidence interval
State: Parameter of interest and at what confidence level.
Plan: Name inference procedure being used and check conditions.
Do: Actually do the calculations and create the interval.
Conclude: Interpret the interval in context.
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