Estimation

The ASTA team

Point and interval estimates

Point and interval estimates

Point estimators: Bias

Point estimators: Consistency

Point estimators: Efficiency

Notation

Confidence Interval

Confidence interval for proportion

Example: Point and interval estimate for proportion

Chile <- read.delim("https://asta.math.aau.dk/datasets?file=Chile.txt")
library(mosaic)
tally( ~ sex, data = Chile)
## sex
##    F    M 
## 1379 1321
tally( ~ sex, data = Chile, format = "prop")
## sex
##         F         M 
## 0.5107407 0.4892593

Example: Confidence intervals for proportion in R

prop.test( ~ sex, data = Chile, correct = FALSE)
## 
##  1-sample proportions test without continuity correction
## 
## data:  Chile$sex  [with success = F]
## X-squared = 1.2459, df = 1, p-value = 0.2643
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.4918835 0.5295675
## sample estimates:
##         p 
## 0.5107407

General confidence intervals for proportion

Example: Chile data

Compute for the Chile data set the 99% and 95%-confidence intervals for the probability that a person is female:

Confidence Interval for mean - normally distributed sample

\(t\)-distribution and \(t\)-score

The expression of the density function is of slightly complicated form and will not be stated here, instead the \(t\)-distribution is plotted below for \(df =1,2,10\) and \(\infty\).

Calculation of \(t\)-score in R

qdist("t", p = 1 - 0.025, df = 4)

## [1] 2.776445

Example: Confidence interval for mean

Ericksen <- read.delim("https://asta.math.aau.dk/datasets?file=Ericksen.txt")
stats <- favstats( ~ crime, data = Ericksen)
stats
##  min Q1 median Q3 max     mean       sd  n missing
##   25 48     55 73 143 63.06061 24.89107 66       0
qdist("t", 1 - 0.025, df = 66 - 1, plot = FALSE)
## [1] 1.997138
t.test( ~ crime, data = Ericksen, conf.level = 0.95)
## 
##  One Sample t-test
## 
## data:  crime
## t = 20.582, df = 65, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  56.94162 69.17960
## sample estimates:
## mean of x 
##  63.06061

Example: Plotting several confidence intervals in R

An experiment was conducted to measure and compare the effectiveness of various feed supplements on the growth rate of chickens. Newly hatched chicks were randomly allocated into six groups, and each group was given a different feed supplement. Their weights in grams after six weeks are given along with feed types.

cwei <- favstats( weight ~ feed, data = chickwts)
se <- cwei$sd / sqrt(cwei$n) # Standard errors
tscore <- qdist("t", p = .975, df = cwei$n - 1, plot = FALSE) # t-scores for 2.5% right tail probability
cwei$lower <- cwei$mean - tscore * se
cwei$upper <- cwei$mean + tscore * se
cwei[, c("feed", "mean", "lower", "upper")]
##        feed     mean    lower    upper
## 1    casein 323.5833 282.6440 364.5226
## 2 horsebean 160.2000 132.5687 187.8313
## 3   linseed 218.7500 185.5610 251.9390
## 4  meatmeal 276.9091 233.3083 320.5099
## 5   soybean 246.4286 215.1754 277.6818
## 6 sunflower 328.9167 297.8875 359.9458
gf_errorbarh(feed ~ lower + upper, data = cwei) %>% 
  gf_point(feed ~ mean)

Determining sample size

Sample size for proportion

Example

Sample size for mean