Interval Estimationfor Means

Notes of STAT6205 by Dr. Fan

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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

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Interval Estimation

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Confidence vs. Probability

The selection of sample is random.

But nothing is random after

we take the sample!

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(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

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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 *

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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.

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Examples/Problems

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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.

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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?

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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.

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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

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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.

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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?

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Example

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Example

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Paired Samples

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