AP Statistics - Confidence Intervals with Means - One Sample

CONFIDENCE
INTERVALS WITH
MEANS (z and t)
...ONE SAMPLE
Frances Coronel

Bell 7

AP Statistics
FVC productions
REVIEW!

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.

Steps to Correctly Make
a Conﬁdence Interval
1. Assumptions

2. Calculations

3. Conclusions
No
statements!

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)

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

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)

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

2. Calculations (t)
Finding t-critical values with the table

You use Table B: t-distributions. Look up confidence
level on bottom and degress of freedom on sides
where df=n-1

Example: 70% confidence when n=12

location highlighted in blue
Table B t distribution critical values
Tail probability p
df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .0005
1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.6
2 .816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.60
3 .765 .978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 12.92
4 .741 .941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610
5 .727 .920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869
6 .718 .906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
7 .711 .896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408
8 .706 .889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041
9 .703 .883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781
10 .700 .879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587
11 .697 .876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437
12 .695 .873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
13 .694 .870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
14 .692 .868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
15 .691 .866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
16 .690 .865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015
17 .689 .863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
18 .688 .862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.611 3.922
19 .688 .861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883
20 .687 .860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
21 .686 .859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819
22 .686 .858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
23 .685 .858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768
24 .685 .857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745
25 .684 .856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725
26 .684 .856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
27 .684 .855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
28 .683 .855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
29 .683 .854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
30 .683 .854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
40 .681 .851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551
50 .679 .849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496
60 .679 .848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460
80 .678 .846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416
100 .677 .845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
1000 .675 .842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
ϱ .674 .841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291
50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%
Confidence level C
Finding t-critical values with
the table

CALC: 2nd > Vars > 4:invT:

Then you type in invT(P,df)

df=n-1

P=critical value+tail area

Example: 90% confidence when n=5

CALC: invT(.90+.05, 5-1)...

so invT(.95,4)= 2.1318 ≈ 2.132
Confidence
Level
Tail Area
80% 0.1
90% 0.05
95% 0.025
98% 0.01
99% 0.005

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

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

Conﬁdence Levels &
Corresponding z-values
Conﬁdence Level z-value
80% 1.282
90% 1.645
95% 1.96
98% 2.326
99% 2.576
In doubt?

This can all be found on the t critical values table that you
receive on your AP Stat exam.
1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.6
2 .816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.60
3 .765 .978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 12.92
4 .741 .941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610
5 .727 .920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869
6 .718 .906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
7 .711 .896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408
8 .706 .889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041
9 .703 .883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781
10 .700 .879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587
11 .697 .876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437
12 .695 .873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
13 .694 .870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
14 .692 .868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
15 .691 .866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
16 .690 .865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015
17 .689 .863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
18 .688 .862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.611 3.922
19 .688 .861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883
20 .687 .860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
21 .686 .859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819
22 .686 .858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
23 .685 .858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768
24 .685 .857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745
25 .684 .856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725
26 .684 .856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
27 .684 .855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
28 .683 .855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
29 .683 .854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
30 .683 .854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
40 .681 .851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551
50 .679 .849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496
60 .679 .848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460
80 .678 .846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416
100 .677 .845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
1000 .675 .842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
ϱ .674 .841 1.036 1.282 1.645 1.960 2.054 2.326 2.576 2.807 3.091 3.291
50% 60% 70% 80% 90% 95% 96% 98% 99% 99.5% 99.8% 99.9%
Confidence level C

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.

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)

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....

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.

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!

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=

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

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.
Slide 13
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AP Statistics - Confidence Intervals with Means - One Sample

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