Download Confidence Intervals and Tests of Hypothesis

Bus 621 Statistics
Lecture 1
Basics of Statistical Inference
Lecture 1
1.
Inference for a single numerical variable
2.
Statement of hypotheses
3.
P-value concept
4.
How to communicate the results of a test
5.
Inference for a single numerical variable and a categorical variable
with 2 categories
6.
Inference for a single categorical variable
7.
Inference for 2 categorical variables
Statistical Methods
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Tutorials
Estimation
Hypothesis
Testing
Estimation Process
Population


Mean, , is
unknown




Sample





Random Sample
Mean


X
=
50

I am 95%
confident that 
is between 40 &
60.
Unknown Population Parameters
Are Estimated
Estimate Population
Parameter...
Mean

with Sample
Statistic
x
Proportion
p
Std. Dev.

s
1 -  2
x1 -x2
Differences
p
Estimation Methods
Estimation
Point
Estimation
Interval
Estimation
Point Estimation
1. Provides a single value
• Based on observations from one sample
2. Gives no information about how close the value is to
the unknown population parameter
3. Example: Sample mean x = 3 is a point
estimate of unknown population mean
Interval Estimation
1. Provides a range of values
•
Based on observations from one sample
2. Gives information about closeness to unknown
population parameter
3. Example: Unknown population mean lies between 50
and 70 with 95% confidence
Confidence Level
1. Probability that the unknown population parameter
falls within interval
2. Denoted (1 – 
•  is probability that parameter is not within
interval
3. Typical values are 99%, 95%, 90%
Intervals & Confidence Level
Sampling Distribution of Sample Mean
_
/2
1- 
/2
 x = 
_
X
(1 – α)% of
intervals
contain μ
α% do not
Large number of intervals
Factors Affecting
Interval Width
1. Data dispersion
More variability = larger width
2. Sample size
Larger sample = smaller width
3. Level of confidence
(1 – ) Higher confidence =
larger width
© 1984-1994 T/Maker Co.
Accurate Confidence Interval
for Mean ( Unknown)
Assumption: Population must be normally distributed
Thinking Challenge
You’re a time study analyst in
manufacturing. You’ve
recorded the following task
times (min.):
3.6, 4.2, 4.0, 3.5, 3.8, 3.1.
What is the 90% confidence
interval estimate of the
population mean task time?
Confidence Interval for a
Mean () with Unknown , Using MegaStat
•
MegaStat does all calculations for you. We can be 90%
confident that the population mean falls between 3.379 and
4.021.
Applications
An example using L1 One sample numerical variable.xlsx
Problem 1: Obtain and interpret a 95% confidence interval for the
population mean for price per square foot for all combinations of
SAD and with/without a pool.
Problem 2: Check to see if these confidence intervals may be
inaccurate by looking at normality/sample size.
Your Turn: Do PS1 problem 1
Statistical Methods
Statistical
Methods
Descriptive
Statistics
Inferential
Statistics
Estimation
Hypothesis
Testing
What’s a Hypothesis?
A belief about a population
parameter
I believe the mean GPA of
this class is 3.5!
• Parameter is
population mean,
proportion, slope
• Must be stated
before analysis
© 1984-1994 T/Maker Co.
Hypothesis Testing
Population


 


I believe the
population mean
age is 50
(hypothesis).

Random
sample
Mean 
X = 20
Reject
hypothesis!
Not close.
How do we Measure “Close”?
1.
2.
If hypothesized  value were really the true mean, there
should be a high probability of obtaining the observed sample
xbar by pure random chance. Call this the p-value
If the p-value is smaller than, say, 5%, we “reject” the
hypothesized value for .
Basic Idea
Sampling Distribution
It is unlikely
that we would
get a sample
mean of this
value ...
... therefore, we
reject the
hypothesis that
 = 50.
... if in fact this were
the population mean
20
 = 50
H0
Sample Means
Naming Null & Alternative
Hypotheses
1.
Null hypothesis, H0 (pronounced H-oh) always has equality
sign: , , or 
2.
Alternative hypothesis, Ha , opposite of null
3.
Ha always has inequality sign: , , or 
4.
Specified as Ha :  , , or  some value
•
Example, Ha:  < 3
Identifying Hypotheses
Example: Test that the population mean is not 3
Steps:
• State the question statistically (  3)
• State the opposite statistically ( = 3)
—
Must be mutually exclusive & exhaustive
• Designate which is alternative hypothesis (  3)
—
Has the , <, or > sign
• Designate which is the null hypothesis ( = 3)
• Called a two-tailed hypothesis because of  in Ha
What Are the Hypotheses?
Is the population average amount of TV
viewing equal to 12 hours?
• State the question statistically:  = 12
• State the opposite statistically:   12
• Select the alternative hypothesis: Ha:   12
• State the null hypothesis: H0:  = 12
• This is a two-tailed test.
What Are the Hypotheses?
Is the population average amount of TV
viewing different from 12 hours?
• State the question statistically:   12
• State the opposite statistically:  = 12
• Select the alternative hypothesis: Ha:   12
• State the null hypothesis: H0:  = 12
• This is a two-tailed test.
What Are the Hypotheses?
Is the average amount spent in the bookstore
greater than $25?
• State the question statistically:   25
• State the opposite statistically:   25
• Select the alternative hypothesis: Ha:   25
• State the null hypothesis: H0:   25
• This is a one-tailed or right-tailed test.
What Are the Hypotheses?
Is the average cost per hat less than $20?
• State the question statistically:   20
• State the opposite statistically:  ≥ 20
• Designate the alternative hypothesis: Ha:   20
• State the null hypothesis: H0:  ≥ 20
• This is a one-tailed or left-tailed test.
Level of Significance
1.
A “tail” probability of the bell curve used to define how
many std. devs. of xbar to judge “closeness” and to compare
p-value against.
2.
Designated  (alpha)
•
Typical values are .01, .05, .10 (.05 is most common)
3.
Selected by researcher, otherwise will be given in a problem
4.
Defines unlikely values of sample statistic if null hypothesis
is true
p-Value Approach
1. Probability of obtaining a test statistic more extreme
( or  than actual sample value, given H0 is true is
called the p-value
2. 1- (p-value) is called the confidence in Ha
3. 1-  is called the required confidence to conclude Ha
4. Used to make a decision between hypotheses
• If confidence in Ha is greater than the required
confidence, conclude Ha otherwise find H0
acceptable.
The Four Steps of a Hypothesis Test
1. State Hypotheses
2. Determine p-value (MegaStat)
3. Make decision based on 1-p =confidence in
Ha
4. Draw conclusion within context of problem
• If confidence in Ha is greater than the
required confidence, conclude Ha
otherwise find H0 acceptable.
t Test for Mean ( Unknown)
Assumption for p-value to be accurate
• Population is normally distributed
• If not normal, take large sample (n  30)
• Or switch to a test for population median such
as Wilcoxon Mann-Whitney test
One-Tailed t Test Example
Is the average capacity of batteries
less than 140 ampere-hours? A
random sample of 20 batteries had a
mean of 138.47 and a standard
deviation of 2.66. Assume a normal
distribution. Test at the .05 level of
significance.
One-Tailed t Test Solution
• H0:  ≥ 140
• Ha:  < 140
•  = .05
• df = 20 - 1 = 19
p-value =.009 (MegaStat)
We can be 99.1% confident that
Conclusion: the population mean is less than
140 and since that exceeds the
requirement of 95% we can
conclude  < 140
One-Tailed t Test
You’re a marketing analyst for WalMart. Wal-Mart had teddy bears on
sale last week. The weekly sales ($00s)
of bears sold in 10 stores was:
8 11 0 4 7 8 10 5 8 3
At the .05 level of significance, is there
evidence that the average bear sales per
store is more than 5 ($00s)?
One-Tailed t Test Solution*
•
•
•
•
•
•
H 0:   5
p-value = .111 from MegaStat
H a:  > 5
Confidence in Ha = 1- .111 or
 = .05
.889
df = 10 - 1 = 9
Required confidence to There is insufficient evidence
conclude Ha is 95%.
that pop. mean is more than 5
since we can be only 88.9%
confident.
One-tailed T-test for a
Mean () with Unknown , Using MegaStat
One-tailed T-test for a Mean () with
Unknown , Using MegaStat
Hypothesis Test: Mean vs. Hypothesized Value
5.0000 hypothesized value
6.4000 mean Sales ($00)
3.3731 std. dev.
1.0667 std. error
10 n
9 df
1.31 t
.1109 p-value (one-tailed, upper)
Two-Tailed t Test
You work for the FTC. A
manufacturer of detergent claims
that the mean weight of detergent
is 3.25 lb. You take a random
sample of 64 containers. You
calculate the sample average to
be 3.238 lb. with a standard
deviation of .117 lb. At the .01
level of significance, is the
manufacturer correct?
3.25 lb.
Two-Tailed t Test Solution*
•
•
•
•
•
H0:  = 3.25
p-value = .208 from MegaStat
Confidence in Ha = 1- .208 or .792
Ha:   3.25
  .01
df  64 - 1 = 63
Need to be 99% confident
to conclude Ha
There is insufficient evidence pop. mean is not
3.25 since we can only be 79.2% confident.
The null hypothesis is acceptable.
Applications
An example using L1 One sample numerical variable.xlsx
Problem 3: Test the hypothesis that the mean price per square foot mean for SAD3Pool
is different than $320 at a level of significance of .05. How does that compare to the
95% confidence interval you calculated in Problem 1. Use a level of significance of .05
in this problem and all that follow.
Problem 4: Use the Wilcoxon signed rank test to test whether the median price per
square foot for SAD2Pool is different than $320.
Problem 5: Test the hypothesis that the mean price per square foot for SAD1NoPool is
less than 350.
Example 6: Use the Wilcoxon signed rank test to test whether the median price per
square foot for SAD1NoPool is less than $350.
Your Turn: Do PS1 problems 2,3,4
Two Independent Populations
Example applications
1. An economist wishes to determine whether there is a
difference in mean family income for households in
two socioeconomic groups.
2. An admissions officer of a small liberal arts college
wants to compare the mean SAT scores of applicants
educated in rural high schools and in urban high
schools.
How can we tell what to use for these situations?
• Both have a numerical variable and a categorical variable (with
2 categories)
• See “Choosing Situation by Data Type”
Comparing Two Independent Means,
μ1 – μ2, assuming  unknown
Assumptions
• Independent, random samples
• Populations are approximately normally distributed
• Population standard deviations are equal
If at least one population is not normal then an
alternative test is to compare population medians
using the Wilcoxon Mann-Whitney test
Hypothesis Test Example
You’re a financial analyst for Charles Schwab. Is there a
difference in dividend yield between stocks listed on the
NYSE and NASDAQ? You collect the following data:
NYSE
NASDAQ
Number
11
15
Mean
3.27
2.53
Std Dev
1.30
1.16
Assuming normal populations,
is there a difference in average
yield ( = .05)?
© 1984-1994 T/Maker Co.
Independent Samples
Hypothesis Test Solution
•
•
•
•
•
.p-value = .1397
H0: 1 - 2 = 0 (1 = 2)
Confidence in Ha
Ha: 1 - 2  0 (1  2)
  .05
Is 1- .1397 = .8603
df  11 + 15 - 2 = 24
Need to be 95% confident
to conclude Ha
There is little evidence of a difference in means since we
can only be 86.03% confident that the pop. means are
different
Two Sample T-test & C.I. for Mean Difference
Assuming Equal Variances, Using MegaStat
Two Sample T-test & C.I. for Mean Difference
Assuming Equal Variances, Using MegaStat
Hypothesis Test: Independent Groups (t-test, pooled variance)
NYSE
3.27
1.3
11
NASDAQ
2.53mean
1.16std. dev.
15n
24 df
0.740 difference (NYSE - NASDAQ)
1.489 pooled variance
1.220 pooled std. dev.
0.484 standard error of difference
0hypothesized difference
1.53 t
.1397 p-value (two-tailed)
-0.260 confidence interval 95.% lower
1.740 confidence interval 95.% upper
1.000 margin of error
Wilcoxon Mann-Whitney test using MegaStat
Wilcoxon - Mann/Whitney
Test
n
32
29
61
Pr/SF
sum of
ranks
1035.5 SAD1Pool
855.5 SAD2Pool
1891 total
expected
992.000value
standard
69.243deviation
z corrected for ties with continuity
0.621correction
.5346 p-value (two-tailed)
H0: Population Medians are equal
H1: Population Medians are not equal
P-value = .5346
We can only be 46.54 % confident of a difference in population medians.
See L1 2 sample tests excel file for this example.
Applications
An example using the L1 2 sample tests.xlsx
excel file.
Example 7: Test whether price per square foot
has the same population means for homes with
and without pools.
Your Turn: Do PS1 problems 5,6,7
A single categorical variable:
Z Test for a Proportion
1. Condition
• nπ and n(1-π) > 5
2. Z-test from MegaStat
Example: Do ranch style homes make up less than 50%
of the population of homes?
Data: A sample of 108 homes revealed that 54 were
ranch style.
One-Tailed Solution
• H0: π ≥ 0.50
• Ha: π < 0.50
•
  = .05
P-value = .0271 from Excel
MegaStat
We can be 97.29% confident that the population
proportion is less than 0.5 and therefore can
conclude that π < 0.50
95% confidence interval estimate for π
We can be 95% confident that the population
proportion falls between .3147 and .5001.
Note: A 2-tailed test would have found the null
hypothesis acceptable.
Applications
An example using the L1 Categorical variables
tests and CI-1.xls file.
Example 8: Test whether less than 50% of the
homes are ranch style in the population and
obtaining a 95% interval estimate for that
population proportion.
Your Turn: Do PS1 problems 8
Two categorical variables:
Chi-square Test for Independence
1. Chi-square test statistic
Example: Do 3 different school districts have the same
percentage of ranch, trilevel and two-story homes?
Data: A sample of 108 homes revealed the following
table.
Count of
STYLE
STYLE
SAD
SAD1
SAD2
SAD3
Grand Total
Grand
ranch
trilevel twostory Total
8
24
11
15
11
7
21
4
7
44
39
25
43
33
32
108
Chi-square solution
• H0: No relationship
• Ha: Relationship exists
•
  = .05
P-value = .0005 from
Excel MegaStat
We can be 99.95% confident that there is a
relationship between school district and style of
home
A follow up analysis suggests that SAD 1 has
fewer ranch homes and more trilevel homes than
expected and that the reverse holds for SAD 3.
See the Results tab in L1 Categorical variables
file for details.
Applications
An example using the L1 Categorical variables
tests and CI-1.xlsx file.
Example 9: Test whether there is a relationship
between SAD and style of home.
Your Turn: Do PS1 problems 9