Inference Basics
x for samples of size n :
m  2 x
m
m  2 x
Since about
of the samples are
if we create intervals based on a sample mean, x, and
go up and down by
, then
of the
time, we’ll create an interval that
.
We say “we are
and
confident that m lies between
”.
1
Confidence Interval (CI)
 Level of confidence: with repeated samples the probability
the interval will contain the true parameter value.
 Step 1: find an estimate for the parameter (the statistic)
 Step 2: find the margin of error (creating a range of values)
Three conditions: SRS, Normal dist., and σ is known.
Estimate for mean:
Estimate for margin of error :
2
CI – on the calculator
Confidence Interval: Estimate ± margin of error
 x  (so many ) x
xz
*

n
Given data, need to enter: , List location, C-Level
Given stats, need to enter: , x, n, C-Level
On calculator: STAT, TESTS, 7:ZInterval
Select input (Data or Stats), enter appropriate info, then
Calculate
3
Ex 1 . My jogging times for a 3 mile loop around campus
has a known  = 2.7 min. In a random sample of 90 of
these recorded times, the mean time was 22.6 minutes.
Find a 95% C.I. for m.
m 
Given stats, need to enter: 2.7, x=22.6, n=90, C-Level=.95
 z - Interval :
 margin of error :
 We conclude
4
Ex 2 . A certain breed of hummingbirds is being studied in
southeast GA. A small group of 15 are tagged and weighed.
Based on past studies, we assume weights are Normally
distributed with  = 1.1 g. Find a 90% C.I. for m with
Weights = 2, 2, 2, 2, 2, 2, 3, 3, 3.25, 3.5, 4, 4, 4.5, 5, 5
m 
Given data, need to enter: data in List, 1.1, C-Level=.90
 z - Interval :
 margin of error :
 We conclude
5
Ex 3 . The average high temperature in November in
Savannah for the past 40 years averaged 71.16 degrees.
Assume average high temperatures are Normally
distributed with  = 3° F. Find 90% and 95 % CI.
m 
Given stats, need to enter: 3, x=71.16, n=40, C-Level=.90
 z - Interval :
 margin of error :
 We conclude
6
Ex 3 – Continued – Find 90% and 95 % CI.
m 
Given stats, need to enter: 3, x=71.16, n=40, C-Level=.95
 z - Interval :
 margin of error :
 We conclude
 Result of increasing confidence :
7
Ex 4 . A sample of 20 pumpkins averaged 9.2 pounds.
Assume weights are Normally distributed with  = 1.5 lb.
Find a 92% CI.
m 
Given stats, need to enter: 1.5, x=9.2, n=20, C-Level=.92
 z - Interval? :
8
Significance Tests
 Someone makes a claim that you do not believe.
 So you look for evidence against the claim (supporting your
belief).
 If the claim were true, then how likely would it be to see a
random sample behave the way it did?
 Assume parameter (mean) is distributed Normally
-3
-2
-1
0
1
2
3
9
Significance Tests – Steps
 State parameter being tested
 State hypotheses: H0, the null hypothesis, usually no effect
Ha, the alternative, claim for which you are
trying to find evidence to support
 Compute test statistic: if the null hypothesis is true, where
does the sample fall? Test stat = z-score
 Compute p-value: what is the probability of seeing a test stat
as extreme (or more extreme) as that?
 Conclusion: small p-values lead to strong evidence against H0.
10
Significance Tests – Hypotheses
 H0, the null hypothesis, usually no effect.
 Ha, the alternative, claim for which you are trying to find
evidence to support.
 Three kinds of alternatives:
11
Levels of Significance, α
α = .005
α = .01
α = .05
α = .10
Perform a significance test and get p-value of .037:
12
ST – on the calculator
On calculator: STAT, TESTS, 1:Z –Test
Given data, need to enter: m0, , List location, Ha
Given stats, need to enter: m0, , x, n, Ha
Select input (Data or Stats), enter appropriate info, then
Calculate or Draw
Output: Test stat, p-value
13
Ex 5. Nationally, about 11% of the wheat crop is destroyed by
hail. An insurance company investigates whether GA crops
suffered damage different from the national average, N(11, 5).
16 GA claims of damage had a mean of 12.5% crop damage.
m 

H0 :
-3 -2 -1 0
Ha :
 Test stat : z 
1 2
3
 p  value :
 conclusion :
14
Ex 6. A car manufacturer advertises a new car that gets 47 mpg.
You suspect the manufacturer is exaggerating the mileage. A
sample of 20 cars were tested and found to have a mean mpg of
45.2 miles per gallon. If  = 2.7, is there evidence at the 1%
significance level that the manufacturer is overstating mpg?
m 

H0 :
Ha :
 Test stat : z 
-3 -2 -1 0
1 2
3
 p  value :
 conclusion :
15
Ex 7. The mean running time for a certain type of battery has been
9.8 hours. The manufacturer has introduced a change in the
production method and wants to perform a test to determine
whether the mean running time has increased. Assume  = 2.1
hours and the sample mean of 40 batteries was 10.6 hours.
m 

H0 :
Ha :
 Test stat : z 
-3 -2 -1 0
1 2
3
 p  value :
 conclusion :
16
Ex 8. A laboratory tested 12 chicken eggs from a local farm and
found the mean amount of cholesterol was 230 mg. You believe
this is significantly lower than the stated mean value for
cholesterol in eggs, 240 mg, with  = 19.9 mg. Test your claim.
data= 200, 200, 210, 220, 230, 235, 235, 240, 240, 245, 250, 255
m 

H0 :
Ha :
 Test stat : z 
-3 -2 -1 0
1 2
3
 p  value :
 conclusion :
17
More with Inference
Three conditions: SRS, Normal dist., and σ is known.
Confidence Interval: Estimate ± margin of error
 x  (so many ) x
xz
*

n
Significance Test: m  population mean being tested
H 0 : m  m0
Test Stat : z 
x  m0
x
x  m0

/ n
18
Changing margin of error
Margin of error = z
*

n
Decrease margin of error by:
19
Deciding on sample size
If you want a margin of error at a certain level, m,
what sample size is needed for a given confidence level?
Given :  , m, z
*
Margin of error =
20
Deciding on z*
If you want a given confidence level, how do you get z*?
-3 -2 -1 0
-3 -2 -1 0
1 2
1 2
3
3
21
Example
A 99% C.I. is [18.8, 48.0] for the mean duration of
imprisonment in months.
(a) What is the margin of error?
(b) What does that say about estimating mean duration?
(c) What minimum sample size is needed if you want a margin of error
of at most 12 months (with 99% confidence and σ = 35 months)?
A sample size of
is required.
22
Example-continued
(d) Create a 99% C.I. from a random sample of 57 prisoners.
Given stats, need to enter:
  35, x = 34.2, n = 57, C-Level=.99
 z - Interval :
 margin of error :
23
Ex 3 – Find 95 % CI for temperature.
 m  mean average high temp in Nov in Savannah
Given stats: 3, x=71.16, n=40, C-Level=.95
 z - Interval : (70.23, 72.09)
 margin of error : 71.16  70.23  .93
What minimum sample size is needed if you want a margin of
error of at most .75 degrees with 95% confidence?
A sample size of
is required.
24
Error in Significance Tests
Type I: If we reject H0 when, in fact, H0 is true.
Type II: If we fail to reject H0, when, in fact, Ha is true.
Perform a test: H0: μ = 4; Ha: μ >4
If evidence says to reject H0, and μ=4, then
If evidence says to reject H0, and μ>4, then
If evidence says to not reject H0, and μ=4, then
If evidence says to not reject H0, and μ>4, then
25
Ex 7 – The mean running time for a certain type of battery has
been 9.8 hours. The manufacturer has introduced a change in
production and wants to perform a test to determine whether the
mean running time has increased. Assume  = 2.1 hours and the
sample mean of 40 batteries was 10.6 hours.
 m  mean running time of new battery

H 0 : m  9 .8
H a : m  9.8
 Test stat : z  2.41  p  value :  .0080
 conclusion : There is strong evidence to reject H0 .
If, in fact, the mean running time is equal to 10.1, then your
conclusion would be classified as a:
Type I error
Type II error
correct decision
26
Ex 8 – A laboratory tested 12 chicken eggs from a local farm and
found the mean amount of cholesterol was 230 mg. You believe
this is significantly lower than the stated mean value for
cholesterol in eggs, 240 mg, with  = 19.9 mg. Test your claim.
 m  mean cholestero l level in eggs
H 0 : m  240
 Test stat : z 

H a : m  240
 1.74  p  value :  .0409
 conclusion : There is enough evidence to reject H0 .
If, in fact, the mean cholesterol level is equal to 240, then your
conclusion would be classified as a:
Type I error
Type II error
correct decision
27
Inference for One Sample Mean
Confidence Interval: Estimate ± margin of error
Significance Test:  State parameter being tested
 State hypotheses
 Compute test statistic
 Compute p-value
 Conclusion
28
Assumptions
Three conditions: SRS, Normal dist., and σ is known.
Now:
Which means
Instead of:
x 
z

is no longer used.
We’ll use:
n
x  m0
x
29
t-distributions
Not quite Normal, still symmetric and bell-shaped,
gets closer to Normal curve as sample size increases
-3
-2
-1
0
1
2
3
30
Confidence Interval: Estimate ± margin of error
 x  (so many )(S.E.)
* s
 x t
n
On calculator: STAT, TESTS, 8:TInterval
Given data, need to enter: List location, C-Level
Given stats, need to enter: x, s, n, C-Level
Select input (Data or Stats), enter appropriate info, then
Calculate
31
Ex 9. An adult patient has been treated for tetany, severe
muscle spasms. This condition is associated with low
levels of calcium, an average less than 6 mg/dl. Based
on 10 recent calcium tests, find a 99.9% C.I. for m.
9.3 8.8 10.1 8.9 9.4 9.8 10.0 9.9 11.2 12.1
m 
 t - Interval :
 margin of error :

32
Ex 10. Drivers along a stretch of Abercorn were randomly
selected to determine average speeds. In a sample of 23
cars, the mean speed was 49 mph and the standard
deviation 4.25 mph. Find a 90% CI.
m 
Given stats, need to enter: x=49, s = 4.25, n=23, C-Level=.90
 t - Interval :
 margin of error :

33
Ex 11. A new process for creating artificial sapphires is
being studied. From a random sample of 37 sapphires,
the mean weight is found to be 6.75 carats with a
standard deviation of .33 carats. Find a 99% CI.
m 
Given stats, need to enter: x=6.75, s = .33, n=37, C-Level=.99
 t - Interval :
 margin of error :

34
Significance Tests
On calculator: STAT, TESTS, 2:T –Test
Given data, need to enter: m0, List location, Ha
Given stats, need to enter: m0, x, s, n, Ha
Select input (Data or Stats), enter appropriate info, then
Calculate or Draw
Output: Test stat, p-value
35
Ex 12. Do Honolulu residents have shorter lifespans than other
Hawaiians? In a sample of 20 Honolulu residents, the mean
lifespan was 71.4 years with a standard deviation of 15.62
years. The average Hawaiian lifespan is 77 years. Perform a
significance test at the 5% level.
m 

H0 :
Ha :
 Test stat : t 
 conclusion :
-3 -2 -1 0
1 2
 p  value :
36
3
Ex 13. Do drivers pay attention to the posted speed limits? In a
sample of 23 cars, the mean speed was 49 mph and the standard
deviation 4.25 mph. Is there evidence at the 10% significance
level that drivers drive at something other than the posted speed
limit of 50 mph?
m 

H0 :
Ha :
 Test stat : t 
 conclusion :
-3 -2 -1 0
1 2
3
 p  value :
37
Ex 14. In producing artificial sapphires, you want to know if a new
method makes something other than the industry’s average
standard of 7 carat gems. From a random sample of 37 sapphires,
the mean weight is found to be 6.75 carats with a standard
deviation of .33 carats. Is there evidence at the a = .01 level?
m 

H0 :
Ha :
 Test stat : t 
-3 -2 -1 0
1 2
3
 p  value :
 conclusion :
38
Ex 10&13
Compare C.I to two-tailed S.T.
m 
stats: x=49, s = 4.25, n=23, C-Level=.90, a = .10
 t - Interval :
 S.T. conclusion :
Ex 11&14
m 
stats: x=6.75, s = .33, n=37, C-Level=.99, a = .01
 t - Interval :
 S.T. conclusion :
39
Ex 15. Do educational toys make a difference? Using 6 pairs of
identical twins to lessen any outside factors, one child is
given educational toys and the other child is given noneducational toys. The difference in reading level is calculated
for each pair. (age for exp. – age for con.) x  -2.44, s  2.16
m 

H0 :
Ha :
 Test stat : t 
 p  value :
 conclusion :
40
Single Population Proportions
1. Asking about categorical variables
2. Questions like: Yes or No? Option 1, 2, or 3?
We want to make an inference for the proportion of a
population that exhibit a certain characteristic.
p = population proportion that has some characteristic
p̂  sample proportion that has the characteristic
An individual in a sample is a success if it has the quality.
p̂ 
41
Sampling distribution for samples of size n
from a population with p.
Mean:
Std. Deviation:
For large values of n:
42
Confidence Interval:
Estimate ± margin of error  p
ˆ  z * S .E.
S .E. 
ˆ (1  p
ˆ)
p
n
This is best with large sized sample and at least 15 of
each, successes and failures.
On calculator: STAT, TESTS, A:1-PropInt…
Enter: x = number of successes
n = sample size
Confidence level
43
Significance Test:
p  population proportion being tested
H 0 : p  p0
Test Stat : z 
 pˆ 
pˆ  p0
 pˆ

p0 (1  p0 )
n
pˆ  p0
p0 (1  p0 )
n
On calculator: STAT, TESTS, 5:1 –PropZTest
need to enter: p0, x, n, Ha
Note: output gives z = (test stat) and p = (p-value)
44
Ex 16. Create a 96% C.I for estimating the proportion of all
escaped convicts who will be eventually recaptured.
Data Summary: n  10351, recaptured  7867
p
 C.I. : estimate  margin of error
 margin of error :

45
Deciding on sample size
If you want a margin of error at a certain level, m,
what sample size is needed for a given confidence level?
Given : m, z
*
where p* is some guess for the sample proportion.
A value of p* = .5 is the most conservative (without any info on
which to base a guess).
What sample size is needed in order to estimate the proportion of
people voting for the Democratic candidate if the margin of
error is to be no larger than 0.03 with a 99% confidence level?
46
Ex 17. Is new method of sight restoration better than an old one
where only 30% of patients recover their sight? Test at the
1% significance level.
Summary: n  225, recovered sight  88


H0 :
 Test stat : z 
Ha :
 p-value

47
Ex 18. Diltiazem causes headaches in 12% of hypertension
patients. Will regular exercise reduce this side effect?
Test at the 1% significance level.
Summary: n  209, headache sufferers  16


H0 :
Ha :
 Test stat : z 
 p-value

48