ECON1203 Lecture Notes - Lecture 9: Confidence Interval, Null Hypothesis, Interval Estimation
8 – Confidence Intervals
Interval estimators
• Produce an interval (i.e. a range of values) and a degree of confidence associated
with that interval
Interval estimation for means
• Suppose X bar ~ N (µ, 2 / n) and thus Z =
• Consider a symmetrical interval around this standardised value of
:
Say we choose = .05 → P (-∞ ≤ Z ≤ bound) = 0.25 → bound = 1.96
P
Can now rearrange this statement to yield:
(a)
The endpoints,
define a confidence interval
• Endpoints of the interval are themselves random variables
We have constructed a random interval
• µ is a constant
• For a specific sample (and sample mean value), µ is either in the confidence interval,
or it is not
If size-n samples were drawn, we would expect 95 of them to include µ
Confidence intervals
• CIs for eas ad proportions typically have a similar structure
Centered at sample statistics
Edpoits are ± soe ultiple of the stadard error if e dot ko siga or
standard deviation (if we do know sigma) of the sampling distribution
The ultiple is determined by the confidence level chosen by the indicator
(a) Remember – if ou dot ko siga ad hae a sall saple, use t-
distribution tables to get bounds (not Z)
Hypothesis testing examples and concepts again
• Maitaied or ull hypothesis
Some statement about a population parameter
• Alternative hypothesis
Depends on research objective
• How data are used to test a null hypothesis
Proceed by comparing a test statistic with the value specified in H0 and decide
whether the difference is:
(a) Small enough to attribute to random sampling errors → do not reject H0
(b) So large that H0 is ore likely ot to e orret → reject H0
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Document Summary
The endpoints, 1. 9(cid:888) define a confidence interval: endpoints of the interval are themselves random variables. Can now rearrange this statement to yield: Interval estimators: produce an interval (i. e. a range of values) and a degree of confidence associated with that interval. We have constructed a random interval: is a constant, for a specific sample (and sample mean value), is either in the confidence interval, or it is not. (cid:858)if (cid:1005)(cid:1004)(cid:1004) size-n samples were drawn, we would expect 95 of them to include (cid:859) Confidence intervals: ci(cid:859)s for (cid:373)ea(cid:374)s a(cid:374)d proportions typically have a similar structure. E(cid:374)dpoi(cid:374)ts are so(cid:373)e (cid:373)ultiple of the sta(cid:374)dard error (cid:894)if (cid:449)e do(cid:374)(cid:859)t k(cid:374)o(cid:449) sig(cid:373)a(cid:895) or standard deviation (if we do know sigma) of the sampling distribution. The (cid:858)(cid:373)ultiple(cid:859) is determined by the confidence level chosen by the indicator (a) remember if (cid:455)ou do(cid:374)(cid:859)t k(cid:374)o(cid:449) sig(cid:373)a a(cid:374)d ha(cid:448)e a s(cid:373)all sa(cid:373)ple, use t- distribution tables to get bounds (not z)
