Chapter 7  μ Confidence Intervals and Sample Size

Chapter 7 Confidence Intervals and Sample Size
7.1 Introduction: Sample statistics are used to
estimate the true population parameters
7.2 Confidence Intervals for the Mean
(σ known or n ≥30) and Sample Size
Point Estimate: a specific numerical values estimate of a
parameter. The best point estimate of the population μ
the sample mean
Sample statistics are call estimators.
good estimators:
1) Unbiased (right on target)
2) Consistent ( approaches true parameter)
3) relatively efficient (smallest variance)
Confidence Intervals
Interval estimator:of a parameter is an interval or
a range of values used to estimate the parameter.
the estimate may or may not contain the value of
the parameter being estimated.
Example: 26.9< μ < 27.7 The true population mean may
lie between the two numbers. The degree of confidence is given with the interval. example: 95% (other common 90% or 99%)
Example Statement: One can be 95%confident that the true population mean lies between 26.9 and 27.7.
Confidence Level of an interval estimate is the probability that the interval estimate will contain the parameter.
Confidence Interval is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.
The z­score associated with the confidence level
is symbolized by zα /2 Where α/2 is the area in each tail of the normal curve. 1­α = .95 for a
95% confidence level.
Formula for the confidence Interval of the Mean
for a specific α Common confidence level z­scores:
90%: zα /2 = 1.645, 95%: zα /2 = 1.96, 99%: zα /2 = 2.575
called the maximum error of estimate.
the maximum error of the estimate is the maximum likely
difference between the point estimate of a parameter and the actual value of the parameter.
Round to at least one more decimal point than the data.
Page 352 Example 7.1
7.1a) Let = $15 and s = 6.5 find the best point
estimate of the population mean and the 90% confidence interval of the population.
Page 352 Example 7­2
7­2a) Find the 90% and 95%confidence intervals for the population mean. Graphs page 353:
Central limit theorem: approximately 95% of the sample means fall with in 1.96 standard deviations
of the population mean. n >30
Find zα /2 for a 98% confidence interval:
Find zα /2 for a 80% confidence interval:
Find zα /2 for a 85% confidence interval:
Page 354 Example 7­3.
Use your calculator to find the 90% confidence intervals for the population mean from the data given.
7.2 Day 2
Sample Size
May want the maximum error of estimate to be within a specific range with a specific confidence level.
since the maximum error estimate
E= Then the formula to determine the sample
size needed is:
Page 356 Example 7­4
7­4a) If using a 95%confidence interval?
b) 95% confident and within 1.5 years?
Assignment: page 358, #1­6, 7­25 odd, #10
7­3 Confidence Intervals for the Mean
(σ Unknown and n<30)
t­distribution is used when σ Unknown and n<30
t­distribution is similar to the normal
1) bell shaped
2) symmetric about mean
3) mean, median, mode=0
4) curve never touches the x­axis
Differs from normal
1) variance is >1
2) a family of curves based on sample size (or df)
3) as size increases t approaches the normal
degrees of freedom
used for different statistics
number of values that are free to vary after a sample statistic is used.
df = n­1
For some statistical tests other than t: df= n­k
t­distribution chart is included on pull out card in book.
It is read differently than the z­chart. t­scores are used in confidence interval for small
samples and where σ is not known
Formula for the confidence interval:
In table find the t­score for:
a) n=15, 95%CI
b) n=15, 99%CI
c) n=15, 90%CI
d) n= 30, 95% CI
e) n =5, 95% CI
f) n= 25, 98%CI
When to use z or t
Is σ known?
Use zα/2 for any size n
Is n ≥ 30?
Use zα/2 and s
Use tα/2 and s
Page 364 Example 7­6
a) find the 90% and 99% confidence intervals
b) what if the sample size was 18, find the 99% CI
Example 7­7 Using your calculator
a) find the 85% CI
One can be 85% confident the true average number of fires started by candles per year is between 6037.1 and 8045.8 Assignment: page 366,# 1­5, 7­19 odd