Download 1 Chapter 7.1: Basic Properties of Confidence Intervals Instructor: Dr

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Chapter 7.1: Basic Properties of Confidence Intervals
Instructor: Dr. Arnab Maity
Outline of this chapter:
• Definition of a confidence interval (CI)
• How to interpret a CI
• Sample size considerations
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Review of Chapter 6
1. Parameter
2. Point estimator
• a random variable
• has a sample distribution
3. Point estimate
4. Properties of interest
• Bias
• Variance and standard error of an estimator
5. Two ways to form an estimator
• Method of Moments Estimators (MOME)
• Maximum Likelihood Estimators (MLE)
– invariance property
A point estimate tells nothing about how close our estimate is to the true parameter.
In fact, the true value will never actually be equal to our estimate. For this reason, we
might prefer to use interval estimates, or a confidence interval, which contains a
set of plausible values of the true parameter.
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• A confidence interval (CI) gives a range of values in which the parameter being
estimated is expected to fall.
• Associated with the interval is a confidence level, which is a measure of the degree
of reliability of the interval. The most frequently used confidence levels are 95%, 99%
and 90%.
Suppose the parameter of interest is θ. We want to estimate θ based on a random
sample. Given a confidence level, say 95%, we wish to find the lower confidence limit
(LCL) and the upper confidence limit (UCL) based on the sample such that
P (LCL ≤ θ ≤ U CL) = 0.95
In other words, a good interval estimator will successfully include the true value 95%
of the time over repeated sampling.
• A good interval will be narrow and have a high confidence level. A narrow confidence interval suggests that our knowledge of the value of the parameter is reasonably
precise, and vice versa.
• Notation of zα :
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Confidence interval for population mean µ (variance σ 2 is known)
Suppose X1 , . . . , Xn is a random sample from a N (µ, σ 2 ) distribution. Assume that σ 2 is
known. We want to form a 95% confidence interval of µ
Step 1: We know that sample mean X̄ is an unbiased estimator of µ (from Chapter 6).
What is the sampling distribution of X̄?
Step 2: Standardize X̄ (subtract mean and divide by sd). What is the sampling distribution of the standardized random variable?
Step 3: Obtain an interval with confidence level 95%.
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1. Suppose that birth weights for boys are normally distributed. We want to estimate
the population mean µ, the overall average birthweight for boys. Assume that the
population standard deviation σ is known to be 12 oz. A random sample of 16 newborn
boys yields the sample mean x̄ = 122 oz.
(a) What should we use as our point estimate of µ, the overall average birthweight
for boys?
(b) Construct a 95% confidence interval for the overall average birthweight for boys.
(c) Construct a 90% confidence interval for the overall average birthweight for boys.
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Question: Take the 90% confidence interval, (117.05, 126.95), you computed in the
previous page. Is it correct to say the true value of µ is within those two numbers with
probability 90%, that is, can we say that
P (117.05 < µ < 126.95) = 0.90?
Explain your answer.
Interpretation of confidence intervals: It is correct to say that for a 95% confidence
interval, if repeated intervals were constructed in the same manner for different samples,
then 95% of them would contain the true population parameter.
How to construct general 100(1 − α)% CI for a given value of α?
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• Interval width is defined as
σ
w = 2 · zα/2 · √ .
n
Three factors impact the length of a C.I.
(a) Population standard deviation σ
(b) Sample size n
(c) Confidence level (100(1 − α))%
• The general formula for the sample size n necessary to ensure an interval width w is
h
σ i2
.
n = 2zα/2 ·
w
Always round n up.
2. Suppose that birth weights for boys are normally distributed. Assume that the population standard deviation σ is known to be 12 oz. We want to estimate the population
mean µ.
• If the sample size is 16 (as in the previous problem), find the width of the 95%
confidence interval you constructed before.
• Find the sample size required if the width of the 95% confidence interval is 6.
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3. The weight of fish in a lake is distributed with a mean of 2.2 pounds with a standard
deviation of 0.7 pounds. A nearby lake is believed to have similar fish, and the same
standard deviation. A random sample of 44 of the fish in the nearby lake had an
average weight of 1.9 pounds. Assume that the population distribution is normal.
(a) Find a 99% confidence interval for the average weight of fish in the nearby lake.
State the assumptions that are met and interpret the calculated interval in the
context of the problem.
(b) Interpret the meaning of confidence in the context of the above question.
(c) If we wanted 100% confidence in a C.I., what would happen to our interval and
why?
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Upper or Lower Confidence Bounds for µ (variance σ 2 is known)
Sometimes we may only care about creating an upper or lower bound for µ.
• 100(1 − α)% Upper confidence bound for µ is defined as
√
(−∞, X̄ + zα σ/ n)
• 100(1 − α)% Lower confidence bound for µ is defined as
√
(X̄ − zα σ/ n, ∞)
4. In the birth weight problem (Problem 1), Find a 95% upper bound for the overall
average weight for boys. Interpret the bound in the context of the problem.