DSS/Hypothesis_Testing_Worked_Example.Rmd at master · DragonflyStats/DSS · GitHub

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---
title: "Hypothesis Testing and Statistical Concepts"
output: html_document
---

## Question 1: Hypothesis Testing

In a sample of 100 patients, test whether the mean RDI is _x_ or more using a one-sided 5% type I error rate.
Sample mean: 12
Standard deviation: 4
Find the largest value of _x_ such that H₀: µ = x vs Hₐ: µ > x is rejected.

```r
n <- 100
m <- 12
z <- qnorm(0.05)
sd <- 4
alpha <- 0.05

mu0 <- m + z * sd / sqrt(n)
mu0
```

**Answer**: 11.34 (Choice A)

---

## Question 2: Paired T-Test

SBP measured at baseline and after two weeks on medication:

```r
pharm <- data.frame(
  baseline = c(140,138,150,148,135),
  week2 = c(132,135,151,146,130)
)

t.test(pharm$baseline, pharm$week2, alternative = "two.sided", paired = TRUE)
```

**p-value**: 0.08652

---

## Question 3: Confidence Interval

Test H₀: µ = µ₀ vs Hₐ: µ ≠ µ₀ (alpha = 0.05).

```r
n <- 9
x_bar <- 1100
sdev <- 30
a <- 0.05

round(x_bar + c(-1,1) * qt(1 - (a/2), n - 1) * sdev / sqrt(n))
```

**Confidence interval**: 1077 to 1123

---

## Question 4: Exact Binomial Test

3 out of 4 people preferred Coke:

```r
binom.test(x = 3, n = 4, p = 0.5, alternative = "greater")
```

**p-value**: 0.3125

---

## Question 5: Hospital Infection Rate

```r
p <- 1 / 100
p_ <- 10 / 1787
n <- 1787
serror <- sqrt(p * (1 - p) / n)
test_z <- (p - p_) / serror

pnorm(test_z, lower.tail = FALSE)
```

**p-value**: 0.03067

---

## Question 6: Two-Sample T-Test

```r
mean.diff <- -3 - 1
df <- 16
s_tr <- 1.5
s_pb <- 1.8

pooled.var <- (s_tr^2 * 9 + s_pb^2 * 9) / df
se.diff <- sqrt(pooled.var / 9 + pooled.var / 9)

t.obt <- mean.diff / se.diff
p.value <- 2 * pt(t.obt, df = df)
t.obt
p.value
```

**p-value**: 0.0001852

---

## Question 7: Hypothesis Testing from CI

Given 90% CI of (1077, 1123), test H₀: µ = 1078 (α = 0.05, two-sided).
**Conclusion**: Do not reject H₀ as 1078 is within the CI.

---

## Question 8: Power Calculation

```r
power.t.test(n = 100, delta = 0.01, sd = 0.04, type = "one.sample", alt = "one.sided")$power
```

**Power**: ~0.80

---

## Question 9: Sample Size for Power = 90%

```r
power.t.test(power = 0.9, delta = 0.01, sd = 0.04, type = "one.sample", alt = "one.sided")$n
```

**Sample size**: 138.4

---

## Question 10: Power and Alpha

As α increases, **power increases**.

---

## Question 11: Two-Sample Z-Test

```r
mean.1 <- 44
mean.2 <- 42.04
sds <- 12
n <- 288

st.error <- sds * sqrt(1 / n + 1 / n)
z.11 <- (mean.1 - mean.2) / st.error
p.v <- 2 * pnorm(-abs(z.11))

z.11
p.v
```

**p-value**: 0.05

---

## Question 12: Bonferroni Correction for Multiple Testing

```r
FER <- 0.05 / 10
FER
```

**Adjusted alpha**: 0.005