## What is the basic structure of the hypothesis test?

The most common being observation following a normal distribution.

The ᕼo(Null) and Ⱨ(alternative hypothesis are specified) mostly null specifies a particular value of a parameter.

Under the general assumption, we take Ho is true, the distribution of the test statistic is known.

Given the distribution and value of the test statistic and the form of Ⱨ, we can calculate the P-value of the test.

Based on p-value and pre-specified level of significance, we make decision

Fail to reject Ho

Reject the Ho

### One sample t-test in R

we have taken 9 random sample

t.test(x = apt_crisp$weight, mu = 16, alternative = c("less"), conf.level = 0.95)

> apt_crisp = data.frame(weight = c(15.5, 16.2, 16.1, 15.8, 15.6, 16.0, 15.8, 15.9, 16.2))

> x_bar = mean(apt_crisp$weight)

> s = sd(apt_crisp$weight)

> mu_0 = 16

> n = 9

> t = (x_bar - mu_0) / (s / sqrt(n))

> t

[1] -1.2

> pt(t, df = n - 1)

[1] 0.1322336

data: apt_crisp$weight

t = -1.2, df = 8, p-value = 0.1322

alternative hypothesis: true mean is less than 16

95 percent confidence interval:

-Inf 16.05496

sample estimates:

mean of x

15.9

> apt_test_results = t.test(apt_crisp$weight, mu = 16,

+ alternative = c("two.sided"), conf.level = 0.95)

>

> names(apt_test_results)

[1] "statistic" "parameter" "p.value"

[4] "conf.int" "estimate" "null.value"

[7] "stderr" "alternative" "method"

[10] "data.name"

>

> qt(0.975, df = 8)

[1] 2.306004

apt_test_results$conf.int

[1] 15.70783 16.09217

attr(,"conf.level")

[1] 0.95

>

> c(mean(apt_crisp$weight) - qt(0.975, df = 8) * sd(apt_crisp$weight) / sqrt(9),

+ mean(apt_crisp$weight) + qt(0.975, df = 8) * sd(apt_crisp$weight) / sqrt(9))

[1] 15.70783 16.09217