The document discusses how to construct confidence intervals for means using z-scores and t-scores. It outlines the assumptions, calculations, and conclusions for one-sample confidence intervals. The key steps are to check assumptions about the population distribution and sample size, then use the appropriate formula to calculate the confidence interval with either z-critical values if the population standard deviation is known, or t-critical values if the population standard deviation is unknown.
2. Basic Definitions
CI: estimated range of values for a population
parameter calculated from sample data
Confidence Level: number that provides
information on how much “confidence” we have
in the method used to construct a confidence
interval estimate
SO WHY DO WE NEED IT? To estimate an
unknown population parameter.
3. Steps to Correctly Make
a Confidence Interval
1. Assumptions
2. Calculations
3. Conclusions
No
statements!
4. 1. Assumptions (z)
Have an SRS from population
(or randomly assigned treatments)
σ known
Normal (or approx. normal)
distribution
• Given
• Large sample size (n≥30)
5. 1. Assumptions (t)
Have an SRS from population
(or randomly assigned treatments)
σ unknown
Normal (or approx. normal) distribution
• Given
• Large sample size (n≥30)
• Check graph of data
main difference is sigma
another main difference is
that when n is under 30
you must automatically use
t t-test
6. 2. Calculations (z)
In case of z, where the ϭ is known, the formula is:
CI: ⨉ ± z* (ϭ/√n)
Statistic
Critical
Value
Standard
Deviation of
Statistic
Margin of
Error
Confidence Interval: statistic ± z critical value (standard deviation of statistic)
7. 2. Calculations (t)
In case of t, where the ϭ is unknown, the formula is:
Confidence Interval: statistic ± t critical value (standard deviation of statistic)
same as z in
terms of
location of
important terms
9. For the z formula we know...
CI: ⨉ ± z* (ϭ/√n)
1. ⨉ is sample mean from random sample
2. sample size n is large (n≥30)
3. population standard deviation is known
10. For the t formula we
know...
CI: ⨉ ± t* (s/√n)
1. ⨉ is sample mean from random sample
2. sample size n is large (n≥30) OR the
population distribution is normal
3. population standard deviation is unknown
12. 3. Conclusions
We are __ % confident that the true
population mean of ___ context ___ is
between ___ and ___.
You need to know this by memory for
the AP Statistics Exam.
13. 2. Calculations: Using
the Calculator
PROBLEM: We want to develop a 95% confidence interval for the population mean from
a sample size of 35 where we know the sample mean is 100 and the population
deviation is 12. We are going to use a Z-Interval test because sigma is known
CALC: STAT>TESTS>7:ZInterval
Since we know all information, we got to STATS
in ZInterval table (left) and just insert
information where necessary. We then press
Calculate and get interval answer (right)
When you only have the data and not the mean
or n, just go to CALC: STAT>EDIT>L1 and type in
values (left). The process will be the same, you
just press DATA on the Zinterval table (right)
14. EXAMPLE 1: Confidence
Intervals with Means: z
We want to develop a 95%
confidence interval for the population
mean from a sample size of 40
women where we know the sample
mean is 76.3 and the population
deviation is 12.5.
no context in problem
by the way....
15. EXAMPLE 1: Answer
CI: ⨉ ± z* (ϭ/√n)
CI: 76.3 ± 1.960 (12.5/√40)
CI: 76.3 ± 3.87
CI: (72.3, 80.17)
95% confidence goes
with 1.960 z critical
value
CalculationsAssumptions
-SRS
-Normal because
n≥30
-sigma known
Conclusions
We are 95% confident that the true
population mean of women ___ is between
72.3 and 80.17.
16. Determining Sample Size: 1st Option
Problem: 95% confident so 1.96 for critical value z
ϭ is 5.0
CI: ⨉ ± z* (ϭ/√n)
1.96 (5.0/√n)
1.96 (5.0/√n) = 1
5.0/√n = .510
5 = .510 (√n)
9.8 = √n
(9.8)² = (√n)²
96.04 = n
97 = n
Assume Margin of
Error= 1 in order
to solve for n
Margin of
Error
.510=1/1.96
9.8=5/.510
ALWAYS round up!
17. In order to solve for n you must set B, the margin of error, to 1.
This gives you:
B = 1.96 (ϭ/√n) which is just: 1 = 1.96 (ϭ/√n)
The result for solving variable n is:
n= (1.96ϭ/B)² or just n= (1.96ϭ/1)²
which solves n as
n= (1.96(5)/1)²
n=(9.8/1)²
n=96.04 which rounds to 97
Determining Sample Size: 2nd/Easier Option
Problem: 95% confident so 1.96 for critical value z
ϭ is 5.0
Basically the
formula is:
(confidence level)(ϭ)
B( )
2
n=
19. Quiz Answers!
1. A: (299.89, 300.11)
2. A: At 90% confidence
level, z will be 1.645 and
B=1 because we assume the
margin of error is 1 so
n= ((1.645)(9)/(1))² =
(14.805)² n= 219.188
3. D: Assumptions: Have an
SRS from population, σ
known, Normal because
large sample size (n≥30)
35>30 so check
4. B: Zinterval: (5.829,
6.2448)
5. C: CI: ⨉ ± t* (s/√n)
CI: 67.5 ± 1.676 (9.3/√51)
CI: 67.5 ± 2.18258
CI: (65.318, 69.682)
df= 50, s=9.3, t=1.676
Extra Credit: To estimate an
unknown population
parameter
calculated t
critical value
with table with
df as 50 (51-1)
at 90% level,
should get
1.676
20. Sources
" Z - C o n f i d e n ce I nte r va l. " P re n h a l l. N.p. , n . d . We b.
2 3 M a y 2 0 1 1 . < htt p : //w w w.pre n h a l l.co m /e s m /a p p /
ca lc _ v 2/ca lc u lato r/m e d ia li b /Te c h n o lo g y/ D o c u m e nt s /
T I- 8 3/d e s c _ p a g e s /z _ co n f _ i nte r. ht m l >
M a s s ey, Tiffa n y. "C o n f i d e n ce I nte r va l N ote s. " A P
S tat i st ic s: B e l l 7. M H S M at h D e p a r t m e nt. M a u r y
H i g h S c h o o l, N o r fo l k , VA. 2 0 1 0 -2 0 1 1 . L e ct u re s.
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