Upload
marla
View
65
Download
0
Embed Size (px)
DESCRIPTION
Interval Estimation for Means. Notes of STAT6205 by Dr. Fan. Overview. Sections 6.2 and 6.3 Introduction to interval estimation Confidence Intervals for One mean General construction of a confidence interval Confidence Intervals for difference of two means Pair Samples. - PowerPoint PPT Presentation
Citation preview
Interval Estimationfor Means
Notes of STAT6205 by Dr. Fan
6205-ch6 2
Overview• Sections 6.2 and 6.3
• Introduction to interval estimation• Confidence Intervals for One mean• General construction of a confidence interval• Confidence Intervals for difference of two means• Pair Samples
6205-ch6 3
Interval Estimation
6205-ch6 4
Confidence vs. Probability
The selection of sample is random.
But nothing is random after
we take the sample!
6205-ch6 5
(Symmetric) Confidence Interval
• A k% confidence interval (C.I.) for a parameter is an interval of values computed from sample data that includes the parameter k% of time:
Point estimate + multiplier x standard error
• K% of time = k% of all possible samples
6205-ch6 6
Estimation of One Mean mWhen the population distribution is normal
Case 1: the SD s is known Z interval
Case 2: the SD is s unknown t interval
nzX *
nstX *
6205-ch6 7
Estimation of One Mean mWhen the population distribution is not normal but
sample size is larger (n> = 30)
Case 1: the SD s is known Z interval
Case 2: the SD is s unknown Z interval, replacing s by s.
6205-ch6 8
6205-ch6 9
Examples/Problems
6205-ch6 10
Examples/Problems• Example 1: We would like to construct a 95% CI for the true mean weight of a newborn baby. Suppose the weight of a newborn baby follows a normal distribution. Given a random sample of 20 babies, with the sample mean of 8.5 lbs and sample s.d. of 3 lbs, construct such a interval estimate.
6205-ch6 11
Can CI be Asymmetric?• Endpoints can be unequal distance from the
estimate
• Can be one-sided intervalExample: Repeat Example 1 but find its one-sided interval (lower tailed).
• Why symmetric intervals are the best when dealing with the normal or t distribution unless otherwise stated?
6205-ch6 12
How to Construct Good CIs• Wish to get a short interval with high degree of
confidence
Tradeoff:• The wider the interval, the less precise it is• The wider the interval, the more confidence that it
contains the true parameter value.
Best CI:For any given confidence level, it has the shortest interval.
6205-ch6 13
Difference of Two MeansWhen: Two independent random samples from two normal populations
Case 1: variances are knownZ interval
Case 2: variances are unknownwithout equal variance assumption
Approximate t intervalwith equal variance assumption
Pooled t interval
6205-ch6 14
Difference of Two MeansWhen: 2 independent random samples from two non-normal populations but large samples (n1, n2 >= 30)
Case 1: variances are knownZ interval
Case 2: variances are unknownZ interval, replacing si by Si.
6205-ch6 15
6205-ch6 16
Examples/Problems• Example 2: Do basketball players have bigger
feet than football players?
• Example 3: To compare the performance of two sections, a test was given to both sections.
• From an estimation point of view (for variances), why is the pooled method preferred?
• How to check the assumption of equal variance?
6205-ch6 17
Example
6205-ch6 18
Example
6205-ch6 19
Paired Samples
6205-ch6 20