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Interval Estimation & Hypothesis testing
Confidence Intervals
• Confidence intervals for the population mean, μ
– when population standard deviation σ is known
– when population standard deviation σ is unknown
• Confidence intervals for the population proportion, p
• Determining the required sample size

Point and Interval Estimates
• A point estimate is a single number
• A confidence interval provides additional information about variability

Lower
Confidence
Limit

Upper
Confidence
Limit

Point Estimate

Width of
confidence interval
Point Estimates
We can estimate a
Population Parameter …

Mean

with a Sample Statistic (a
Point Estimate)
μ

X Error! Bookmark not

defined
...

Random Sample

Mean X = 50

Sample
Population
(mean, μ, is unknown)

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General Formula
• The general formula for all confidence intervals is:
Point Estimate ± (Critical Value)(Standard Error)

Confidence Level
• Confidence Level
– Confidence for which the interval will contain the unknown population parameter
• A percentage (less than 100%)
Confidence Level, (1- )
• Suppose confidence level = 95%
– Also written (1 - ) = 0
...
08 people per
group
...
398 people
...
03 and 2
...
f
...
f
...
0

Student’s t Distribution

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Confidence Intervals for the Population Proportion, π
• An interval estimate for the population proportion ( π ) can be calculated by adding an
allowance for uncertainty to the sample proportion ( p )

Confidence Intervals for the Population Proportion, π
• Recall that the distribution of the sample proportion is approximately normal if the sample
size is large, with standard deviation

• We will estimate this with sample data
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Confidence Interval Endpoints
• Upper and lower confidence limits for the population proportion are calculated with the

formula

where
Z is the standard normal value for the level of confidence desired
p is the sample proportion
n is the sample size

Example
• A random sample of 268 people shows that 78 are female golfer
• Form a 95% confidence interval for the true proportion of female golfer

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Interpretation
• We are 95% confident that the true percentage of female golfers in the population is between
28
...
22%
...
2897 to 0
...
5)
Required Sample Size Example
How large a sample would be necessary to estimate the true proportion defective in a large
population within 3%, with 95% confidence?

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(Assume a pilot sample yields p = 0
...

A sample of 68 golfers is selected with average score of 75
...
68
Find the 95% confidence interval estimate of total score
...
How can we
check (test) whether it is correct?
•Intuition:
–we try to find evidence to reject or disprove this hypothesis
...

–If we have enough evidence to reject then we believe in the alternative conclusion
...

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What is a
Hypothesis?
•A hypothesis is a statement
(assumption) about a population
parameter
–population mean:
The mean spent on golf game in Sydney
is μ = $285
–population proportion:
The proportion of young golfer in
Sydney is π = 0
...

•In any tests, there are two hypotheses: null and alternative
...

•I believe that more than 25% of the tourist golfers are male

•This research concluded that UK golfers score at least 10 points shorter than US golfer

•US golf clubs operates with minimal profit than UK golf clubs
...
25 against H1: μ > 0
...

•The production process assumes the amount of time to complete a critical part was normally
distributed with a mean of 130 seconds and standard deviation of 15 seconds
...
He measures 100 randomly selected
assemblies and finds the mean is 126
...
Can the supervisor conclude at the 5% level of
significance that the assumption of the mean assembly time being 130 is incorrect?

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