```
V <- 40000
M <- 1300
SD <- sqrt(V)
x <- 979
round(pnorm(x, M, SD, lower.tail = FALSE),4)
```

`## [1] 0.9458`

```
V <- 1960000
M <- 11000
SD <- sqrt(V)
x <- 8340
round(pnorm(x, M, SD, lower.tail = FALSE),4)
```

`## [1] 0.9713`

```
x1 <- 83000000
x2 <- 85000000
M <- 80000000
SD <- 3000000
Px1 <- round(pnorm(x1, M, SD),4)
Px2 <- round(pnorm(x2, M, SD),4)
Px2-Px1
```

`## [1] 0.1109`

```
# Method 1
M = 456
SD = 123
x = 0.14
round(qnorm(0.14, 456, 123, lower.tail = FALSE),0)
```

`## [1] 589`

```
# Method 2
x = 0.86
round(qnorm(0.86, 456, 123),0)
```

`## [1] 589`

```
x1 <- 0.07
x2 <- 0.93
M <- 6.13
SD <- 0.06
round(qnorm(x1,M,SD),2)
```

`## [1] 6.04`

`round(qnorm(x2,M,SD),2)`

`## [1] 6.22`

```
x1 <- 0.2
x2 <- 0.45
M <- 78.8
SD <- 9.8
round(qnorm(x1,M,SD),0)
```

`## [1] 71`

`round(qnorm(x2,M,SD),0)`

`## [1] 78`

```
x <- 0.45
M <- 21.2
SD <- 5.4
round(qnorm(x,M,SD,lower.tail = FALSE),0)
```

`## [1] 22`

```
x <- 11
N <- 151
pi <- 0.09
M <- (N*pi)
SD <- sqrt(M*(1-pi))
# use normal distribution
round(pnorm(x, M, SD, lower.tail = TRUE),4)
```

`## [1] 0.2307`

```
M <- 48
N <- 147
SD <- 7
SEM <- SD/sqrt(N)
round(pnorm(48.83, M, SEM, lower.tail = FALSE),4)
```

`## [1] 0.0753`

```
M <- 91
SD <- 10
N <- 68
SEM <- SD/sqrt(N)
round(pnorm(93.54, M, SEM, lower.tail = FALSE),4)
```

`## [1] 0.0181`

```
N <- 540
M <- 0.07
SE <- sqrt((0.07)*(0.93)/(540))
round(pnorm(0.10, M, SE, lower.tail = TRUE),4)-round(pnorm(0.04, M, SE, lower.tail = TRUE),4)
```

`## [1] 0.9938`

```
N = 602
M = 0.23
SE = sqrt((0.23)*(0.77)/602)
round(pnorm(0.27, M, SE, lower.tail = FALSE) + pnorm(0.19, M, SE, lower.tail = TRUE),4)
```

`## [1] 0.0197`

```
N <- 208
x_bar <- 3.9
sigma <- 0.8
z <- 1.282
# Lower Bound
round(x_bar - z*(sigma/sqrt(N)),1 )
```

`## [1] 3.8`

```
# Upper Bound
round(x_bar + z*(sigma/sqrt(N)),1 )
```

`## [1] 4`

```
N <- 7472
x_bar <- 16.6
sigma <- 11
z <- 2.326
# Lower Bound
round(x_bar - z*(sigma/sqrt(N)),1 )
```

`## [1] 16.3`

```
# Upper Bound
round(x_bar + z*(sigma/sqrt(N)),1 )
```

`## [1] 16.9`

```
t = 0.05 # one sided
df = 26
round(qt(t, df),4)
```

`## [1] -1.7056`

```
sample <- c(383.6, 347.1, 371.9, 347.6, 325.8, 337)
l <- length(sample)
smean <- round(sum(sample)/l, 2)
smean
```

`## [1] 352.17`

```
ssd <- round(sd(sample),2)
ssd
```

`## [1] 21.68`

```
cv <- round(abs(qt(0.10/2, l-1)),4)
cv
```

`## [1] 2.015`

```
# calculate margin of erro
E <- cv*(ssd/(sqrt(l)))
# find lower bound
lower <- round(smean - E, 2)
lower
```

`## [1] 334.34`

```
# find upper bound
upper <- round(smean + E, 2)
upper
```

`## [1] 370`

```
N <- 16
cv <- round(abs(qt(0.20/2, N-1)),3)
cv
```

`## [1] 1.341`

```
# Margin of Error (E)
E <- cv * (SD = 2.45/sqrt(N))
M = 46.4
lower <- round((M - E),1)
lower
```

`## [1] 45.6`

```
upper <- round((M + E),1)
upper
```

`## [1] 47.2`

```
z <- abs(qnorm(0.005,mean=0, sd = 1))
E <- 0.13
SD <- 1.9
ceiling((z^2 * SD^2)/E^2)
```

`## [1] 1418`

```
M <- 12.6
V <- 3.61
SD <- sqrt(V)
E <- 0.19
z <- abs(qnorm(0.025, M, SD))
ceiling((z^2 * SD^2)/E^2)
```

`## [1] 7879`

```
N <- 2089
belowRL <- (N-1734)
fraction <- (belowRL/N)
round(fraction, 3)
```

`## [1] 0.17`

```
z <- abs(qnorm(0.01, m=0, sd = 1))
sd <- sqrt((fraction*(1-fraction)) / N)
lower <- round(fraction - (z * sqrt((fraction*(1-fraction))/N)),3)
lower
```

`## [1] 0.151`

```
upper <- round(fraction + (z * sqrt((fraction*(1-fraction))/N)),3)
upper
```

`## [1] 0.189`

```
N <- 474
spills <- 156
fraction <- round(spills/N, 3)
fraction
```

`## [1] 0.329`

```
z <- abs(qnorm(0.025, m =0, sd =1))
N <- 474
sd <- sqrt((fraction*(1-fraction)) / N)
lower <- round(fraction - (z * sqrt((fraction*(1-fraction))/N)),3)
lower
```

`## [1] 0.287`

```
upper <- round(fraction + (z * sqrt((fraction*(1-fraction))/N)),3)
upper
```

`## [1] 0.371`