Inference for two means
STAT 120
The SE for means
The standard error for ˉx is SEˉx=σ√n
where σ is the population SD of your responseThe standard error for ˉx1−ˉx2 is SEˉx1−ˉx2=√σ21n1+σ22n2
But we usually do not know σ !
- Estimate σ with the sample SD s
Central Limit Theorem for means: One sample
The sampling distribution for a sample mean, ˉx, is approximately N(μ,SEˉx)
When is this approximately “good”?
if X∼N(μ,σ) then ˉX∼N(μ,σ/√n)
if X≁N(μ,σ) then ˉX∼N(μ,σ/√n) if n⩾30
Central Limit Theorem for means: Two independent samples:
The sampling distribution for a difference of two independent sample means is approximately N(μ1−μ2,SEˉx1−ˉx2)
When is this approximately “good”?
- need both n1 and n2 samples sizes big enough for the one-sample condition
Academic Performance Index (API)
Academic Performance Index (API) is a number reflecting a school’s performance on a statewide standardized test
- simple random sample of n=200 schools
- variable
growthmeasures the growth in API from 1999 to 2000 (API 2000 - API 1999).
Academic Performance Index (API)
API
Hypothesis Test
Can we use t-inference methods to compare mean growths?
Both samples sizes (98 and 102) can be deemed large
No severe skewness (but two extreme outliers)
Estimated Standard Error : SEˉxh−ˉxl=√28.753802102+29.95048298=4.1544
Test statistics: t=(25.24510−38.82653)−04.154404=−3.2692
The observed mean difference is 3.3 SEs below the hypothesized mean difference of 0
Two-sample t-test
Welch Two Sample t-test
data: growth by wealth
t = -3.2692, df = 196.71, p-value = 0.001273
alternative hypothesis: true difference in means between group high and group low is not equal to 0
95 percent confidence interval:
-21.774321 -5.388544
sample estimates:
mean in group high mean in group low
25.24510 38.82653 The p-value is 0.001273. If there is no difference between mean growth in the two populations, then there is just a 0.13% chance of seeing a sample mean difference that is 3.27 standard errors or more away from 0.
Outliers
cds stype name sname snum
1 5.471911e+13 E Lincoln Element Lincoln Elementary 5873
2 1.975342e+13 E Washington Elem Washington Elementary 2543
dname dnum cname cnum flag pcttest api00 api99 target
1 Exeter Union Elementary 226 Tulare 53 NA 98 693 504 15
2 Redondo Beach Unified 585 Los Angeles 18 NA 100 745 615 9
growth sch.wide comp.imp both awards meals ell yr.rnd mobility acs.k3 acs.46
1 189 Yes Yes Yes Yes 50 18 <NA> 9 18 NA
2 130 Yes Yes Yes Yes 41 20 <NA> 16 19 30
acs.core pct.resp not.hsg hsg some.col col.grad grad.sch avg.ed full emer
1 NA 93 28 23 27 14 8 2.51 91 9
2 NA 81 11 26 32 16 16 2.99 100 3
enroll api.stu pw fpc wealth
1 196 177 30.97 6194 high
2 391 313 30.97 6194 hight.test with outliers removed
Welch Two Sample t-test
data: growth by wealth
t = -4.395, df = 174.97, p-value = 1.916e-05
alternative hypothesis: true difference in means between group high and group low is not equal to 0
95 percent confidence interval:
-23.571116 -8.961945
sample estimates:
mean in group high mean in group low
22.56000 38.82653 How does removing outliers influence t-test stat and p-value?
Confidence Interval
95% Confidence Interval from the output:
- Without Outliers: (-23.57, -8.96)
- With Outliers: (-21.77, -5.39)
Removing Outliers:
- the difference in means shifted further away from 0
- CI shifted further from a difference of 0
- decrease the SE of our sample difference
Interpretation: We are 95% confident that the mean API growth between 1999 and 2000 for all low wealth schools is anywhere from 8.96 points to 23.57 points higher than the mean API growth for all high wealth schools in California.
Paired Data
Data are paired if the data being compared consists of paired data values. Common paired data examples:
- Two measurements on each case
- natural pairs (twins, spouses, etc)
Use paired data to reduce natural variation in the response when comparing the two groups/treatments
- comparing group 1 and 2 responses among similar individuals
- reduces the effects of confounding variables
- reduces the SE for the mean difference!
Analyzing paired data
Look at the difference between responses for each unit (pair) di=x1,i−x2,i
Analyze the mean of these differences rather than the average difference between two groups sample mean difference: ˉd sample SD of difference: sd population mean difference: μd
Use one sample inference methods for these differences
Tuition example
How much higher is non-resident tuition, on average, compared to resident tuition? Use the Tuition2006.csv lab manual data
- the variable
Diffcomputes the differenceRes-NonRes
tuition <- read.csv("http://math.carleton.edu/Stats215/RLabManual/Tuition2006.csv")
head(tuition)
## X Institution Res NonRes Diff
## 1 1 Univ of Akron (OH) 4200 8800 -4600
## 2 2 Athens State (AL) 1900 3600 -1700
## 3 3 Ball State (IN) 3400 8600 -5200
## 4 4 Bloomsburg U (PA) 3200 7000 -3800
## 5 5 UC Irvine (CA) 3400 12700 -9300
## 6 6 Central State (OH) 2600 5700 -3100Tuition example: Histogram and QQ-plot
Tuition example: t-test
# alternate method
t.test(tuition$Res, tuition$NonRes, paired = TRUE)
##
## Paired t-test
##
## data: tuition$Res and tuition$NonRes
## t = -7.5349, df = 18, p-value = 5.69e-07
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
## -4583.580 -2584.841
## sample estimates:
## mean difference
## -3584.211We are 95% confident that the mean tuition for non-residents is $2585 to $4584 higher than mean tuition for residents
Group Activity 1
- Please download the Class-Activity-21 template from moodle and go to class helper web page
30:00 