```
set.seed(19)
<- sample(c("H", "T"),
coin_flips size = 1000,
replace = T,
prob = c(0.5, 0.5))
```

# Binomial Test

The statistical test used to determine whether the proportion in a binary outcome experiment is equal to a specific value. It is appropriate when we have a small sample size and want to test the success probability \(p\) against a hypothesized value \(p_0\).

## Creating a sample dataset

We will generate a dataset where we record the outcomes of 1000 coin flips.

We will use the

`binom.test`

function to test if the proportion of heads is significantly different from 0.5.

Now, we will count the heads and tails and summarize the data.

```
<- sum(coin_flips == "H")
heads_count <- sum(coin_flips == "T")
tails_count <- length(coin_flips) total_flips
```

` heads_count`

`[1] 513`

` tails_count`

`[1] 487`

` total_flips`

`[1] 1000`

## Conducting Binomial Test

```
<- binom.test(heads_count, total_flips, p = 0.5)
binom_test_result binom_test_result
```

```
Exact binomial test
data: heads_count and total_flips
number of successes = 513, number of trials = 1000, p-value = 0.4292
alternative hypothesis: true probability of success is not equal to 0.5
95 percent confidence interval:
0.4815213 0.5444020
sample estimates:
probability of success
0.513
```

### Results:

The output has a p-value 0.4292098 \(> 0.05\) (chosen level of significance). Hence, we fail to reject the null hypothesis and conclude that the **coin is fair**.

# Example of Clinical Trial Data

We load the `lung`

dataset from `survival`

package. We want to test if the proportion of patients with survival status 1 (dead) is significantly different from a hypothesized proportion (e.g. 50%)

We will calculate number of deaths and total number of patients.

```
library(survival)
attach(lung)
<- sum(lung$status == 1)
num_deaths <- nrow(lung) total_pat
```

` num_deaths`

`[1] 63`

` total_pat`

`[1] 228`

## Conduct the Binomial Test

We will conduct the Binomial test and hypothesize that the proportin of death should be 19%.

```
<- binom.test(num_deaths, total_pat, p = 0.19)
binom_test binom_test
```

```
Exact binomial test
data: num_deaths and total_pat
number of successes = 63, number of trials = 228, p-value = 0.001683
alternative hypothesis: true probability of success is not equal to 0.19
95 percent confidence interval:
0.2193322 0.3392187
sample estimates:
probability of success
0.2763158
```

## Results:

The output has a p-value 0.0016829 \(< 0.05\) (chosen level of significance). Hence, we reject the null hypothesis and conclude that **the propotion of death is significantly different from 19%**.