BIO 201 Homework 5

## Settings for RMarkdown http://yihui.name/knitr/options#chunk_options
opts_chunk$set(comment = "", warning = FALSE, message = FALSE, tidy = FALSE, 
    echo = TRUE, fig.width = 7, fig.height = 7)
options(width = 116, scipen = 10)

setwd("~/statistics/bio201/")

library(ggplot2)

References

1. 7.7-7.9

## 7.7
power.t.test(n = 100, delta = 0.1, sd = 0.54, type = "one.sample")

     One-sample t test power calculation 

              n = 100
          delta = 0.1
             sd = 0.54
      sig.level = 0.05
          power = 0.4498
    alternative = two.sided


## 7.8
power.t.test(n = 100, delta = 0.2, sd = 0.54, type = "one.sample")

     One-sample t test power calculation 

              n = 100
          delta = 0.2
             sd = 0.54
      sig.level = 0.05
          power = 0.9561
    alternative = two.sided


## 7.9
power.t.test(n = NULL, delta = 0.1, sd = 0.54, power = 0.8, type = "one.sample")

     One-sample t test power calculation 

              n = 230.8
          delta = 0.1
             sd = 0.54
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

2. 7.23-7.25

## 7.23
library(BSDA)
res.t <- tsum.test(mean.x = 175, s.x = 35, n.x = 24, mu = 230, var.equal = TRUE)
res.t$p.value
[1] 0.0000000824

## 7.24
dat.mean <- 175
dat.sd   <- 35
dat.n    <- 24
dat.se   <- dat.sd / sqrt(dat.n)
dat.qt   <- qt(p = c(0.025, 0.975), df = dat.n - 1)

dat.mean + dat.qt * dat.se
[1] 160.2 189.8

res.t

    One-sample t-Test

data:  Summarized x 
t = -7.698, df = 23, p-value = 0.0000000824
alternative hypothesis: true mean is not equal to 230 
95 percent confidence interval:
 160.2 189.8 
sample estimates:
mean of x 
      175 


## 7.25: Not done

3. 7.51-7.53

## 7.51: One sample t-test
## 7.52
tsum.test(mean.x = 2.65, s.x = 0.11 * sqrt(20), n.x = 20, mu = 2.88, var.equal = TRUE)

    One-sample t-Test

data:  Summarized x 
t = -2.091, df = 19, p-value = 0.05021
alternative hypothesis: true mean is not equal to 2.88 
95 percent confidence interval:
 2.42 2.88 
sample estimates:
mean of x 
     2.65 

## 7.53
power.t.test(power = 0.8, delta = 0.2, sd = 0.11 * sqrt(20), type = "one")

     One-sample t test power calculation 

              n = 49.44
          delta = 0.2
             sd = 0.4919
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

4. 7.73-7.74

## 7.73
library(foreign)
bone <- read.dta("BONEDEN.DAT.dta")
t.test(with(bone, fn2 - fn1))

    One Sample t-test

data:  with(bone, fn2 - fn1) 
t = -0.0503, df = 40, p-value = 0.9601
alternative hypothesis: true mean is not equal to 0 
95 percent confidence interval:
 -0.03013  0.02866 
sample estimates:
 mean of x 
-0.0007317 

## 7.74
t.test(with(bone, fs2 - fs1))

    One Sample t-test

data:  with(bone, fs2 - fs1) 
t = -1.701, df = 40, p-value = 0.09663
alternative hypothesis: true mean is not equal to 0 
95 percent confidence interval:
 -0.066704  0.005728 
sample estimates:
mean of x 
 -0.03049 

5. 7.93-7.96

iop <- read.table(header = TRUE, text = "
pt.num  IOPab   IOPc
1       18.0    14.5
2       16.0    18.0
3       17.0    11.5
4       18.0    18.0
5       20.0    21.0
6       19.0    22.0
7       19.0    24.0
8       12.0    14.0
9       17.0    16.0
10      21.5    19.0
")

iop
   pt.num IOPab IOPc
1       1  18.0 14.5
2       2  16.0 18.0
3       3  17.0 11.5
4       4  18.0 18.0
5       5  20.0 21.0
6       6  19.0 22.0
7       7  19.0 24.0
8       8  12.0 14.0
9       9  17.0 16.0
10     10  21.5 19.0

## 7.93
## Paired t-test or one sample t-test for difference

## 7.94
## Paired t-test
with(iop, t.test(IOPab, IOPc, paired = TRUE))

    Paired t-test

data:  IOPab and IOPc 
t = -0.0493, df = 9, p-value = 0.9618
alternative hypothesis: true difference in means is not equal to 0 
95 percent confidence interval:
 -2.346  2.246 
sample estimates:
mean of the differences 
                  -0.05 


## One-sample t-test for IOPa+b - IOPc
with(iop, t.test(IOPab - IOPc, var.equal = T))

    One Sample t-test

data:  IOPab - IOPc 
t = -0.0493, df = 9, p-value = 0.9618
alternative hypothesis: true mean is not equal to 0 
95 percent confidence interval:
 -2.346  2.246 
sample estimates:
mean of x 
    -0.05 



## 7.95
## Not the same as accepting null. Decide on two-sided margins of equivalence and see if the confidence interval sits within the margins (difference -2 to +2 in this case).

## 7.96: The confidence interval is not contained in the -2 to +2 margins, thus equivalence is not shown.
with(iop, t.test(IOPab - IOPc, var.equal = T))$conf
[1] -2.346  2.246
attr(,"conf.level")
[1] 0.95