Statistical Sciences 2244A/B Lecture Notes - Lecture 8: Normal Distribution, Confidence Interval, Sampling Distribution
22 May 2018
School
Department
Professor
Stats 2244
Confidence Intervals I
Interpretation, z Cl for mu (sigma known)
Consider the following (approx.) Normal distribution of individual weights (in lbs.) If you randomly
selected one individual from this population, which of the following values would you most expect to
get?
a) 100 lbs
b) 115 lbs
c) 150 lbs
d) 230 lbs
Consider the following (approx.) Normal distribution of mean weights (in lbs), for SRSs of n=15. If you
took a SRS of 15 individuals, what mean value would you most expect to get?
a) 100 lbs
b) 115 lbs
c) 150 lbs
d) 230 lbs
Suppose you dont know the actual sampling distribution (other than it is normal); that is, the mean μ
is unknown. You have your SRS, n=4, with ҧ = 10 cm and you know that sigma = 1 cm. Which of the
following statements could you confidently make?
a) All sample means are equally likely to be exactly μ, as they are to be ≥ standard deviations from
μ.
b) ~5% of the sample means possible will have values more extreme than approximately μ ± ·
cm).
c) ~5% of the samples means possible will have values more extreme than approximately μ ±
2·(0.5 cm)
What values of Z enclose the central 95% of the values from the standard normal
distribution?
a) ±1.00
b) ±1.64
c) ± 1.96
d) ± 2.00
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Suppose you dont know the actual sampling distribution other than it is normal;that is, the mean μ
is unknown. You know the population has a standard deviation of sigma, and you will take an SRS of n,
with resulting mean ҧ. So, were speaking in generalities. If you calculated , how
frequently would the interval you construct include the value of ?
a) 50% of the intervals
b) 90% of the intervals
c) 95% of the intervals
d) cant make this prediction
Simulation Results
- biggest problem with confidence intervals is that you never know which one you have
o you dont know if you have a good C) the black one or the red one
- point of C): use sample data to estimate a parameter bc we dont know what the parameter
actually is)
o the confidence interval is our estimate of what the parameter is
o we never know if the interval we construct, has the parameter in it or it doesnt so you
cant say with certainty that you know the parameter) but you can say if you kept doing
it, 95% of the intervals will include the parameter
z Confidence Interval for mu
- recall: the whole point of a CI is to estimate the population parameter
- first CI we are creating is the z confidence
- goal: take a sampel and build an interval estimate for the population mean
- point estimate is a general term, but there are diff types of point estimates (sample mean,
standard deviation etc)
o ex: dog is a general term, but there are diff types
o the point estimate is always the sample version of the parameter you want
- problem: sample statistics have sampling error
- we dont know how far the sample means is from the real mean
find more resources at oneclass.com
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Document Summary
If you randomly selected one individual from this population, which of the following values would you most expect to get: 100 lbs, 115 lbs, 150 lbs, 230 lbs. Normal distribution of mean weights (in lbs), for srss of n=15. Suppose you don(cid:495)t know the actual sampling distribution (other than it is normal); that is, the mean (cid:523) (cid:524) is unknown. You have your srs, n=4, with = 10 cm and you know that sigma = 1 cm. What values of z enclose the central 95% of the values from the standard normal distribution: 1. 00, 1. 64, 1. 96, 2. 00 is unknown. You know the population has a standard deviation of sigma, and you will take an srs of n, Suppose you don(cid:495)t know the actual sampling distribution (cid:523)other than it is normal(cid:524);that is, the mean (cid:523) (cid:524) with resulting mean . , how: 50% of the intervals, 90% of the intervals, 95% of the intervals, cant make this prediction.
