Download Section 7.5

Section 7.5
Estimating a Population Variance
Symbol Check
= population standard deviation
= population mean
= sample standard deviation
= sample mean
= population variance
= sample variance
Estimators of
• The sample variance
is the best point
estimate of the population variance
.
• The sample standard deviation s is commonly
used as a point estimate of
(even though
it is a biased estimate).
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Constructing Confidence Intervals
Requirements
1.) The sample is a simple random
sample.
2. The population must have a
normal distribution (even if sample
is large).
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Confidence Interval for the Population Variance
R
L
Confidence Interval for the Population Standard Deviation
R
L
Our Distributions
Normal
Distribution
Student t
Distribution
Chi-square
Distribution
Estimates of
proportions or
means with
known
Estimates of
means with
known s.
Estimates of
variance or
standard
deviations.
NEW!
Properties of the Chi-square
Distribution
1. Uses Table A4
▫
To find your Chi-square distribution value you
must know:
 Degrees of Freedom ( df = n-1)
 Area located to the right of the critical value.
Properties of the Chi-square
Distribution
2. Chi-square is NOT symmetric
▫
However, the distribution becomes more
symmetric as the degrees of freedom increase
 Interval Notation: (s²-E < σ <s²+E)
 Interval Notation: (s²-E, s²+E)
 Interval Notation: (s²±E)
Using Table A4
• Find
for the population standard deviation σ
with a confidence level of 95% and a sample size of
n=10.
Constructing a Confidence Interval
• Twelve different video games showing substance use were
observed and the duration of times of game play (in seconds)
are listed below. Use the sample data to construct a 99%
confidence interval estimate of σ, the variance and standard
deviation of the duration of game play.
4049 3884 3859
4027
4318
4813
4657
4823
4334
4317
4033
5004
Determining Sample Size
• We want to be 95% confident that our estimate is within
20% of the true value of σ. Find the sample size.
Homework
• Pg. 377-379
• #2, 6, 7, 16, 19
Homework Answers:
• Pg. 377-379#2, 6, 7, 16, 19
• 2. Yes, no. The format s +/- E cannot be used due to
the lack of symmetry of chi-square distribution. The
confidence interval does not have s at the center.
• 6. 8.907, 32.852
7.) 51.172, 116.321
• 16. 211
• 19. a.) 17.7 min <  < 32.4 min.
b.) 14.7 min <  < 35.3 min
c.) The two confidence intervals are not
substantially different. There does not appear to be a
difference between the standard deviation of lengths
of movies rated PG or PG-13 and the lengths of movies
rated R.