# Hypothesis testing

## Homework Statement

An advertisement for a certain brand of cigarettes claimed that on average there's no more than 18mg of nicotine per cigarettes . A test of 12 cigarettes gave a sample mean of 19.1 . Assuming varience is 4 , test the claim with a significance level of α = 0.05

## The Attempt at a Solution

My ans is
Ho = µ_0 = 18
H1 = µ_0 <18

Since n < 30 , and standard deviation of population unknown , so , i use t-distribution

t test = (19.1-18) / ( 2 /sqrt(2) ) = 1.905

t α = 0.05 , v =11 = 1.796

Since t test > t critical , so i reject the Ho , but accroding to the ans , i should not reject the Ho ,

Is my ans wrong ?

Mentor

## Homework Statement

An advertisement for a certain brand of cigarettes claimed that on average there's no more than 18mg of nicotine per cigarettes . A test of 12 cigarettes gave a sample mean of 19.1 . Assuming varience is 4 , test the claim with a significance level of α = 0.05

## The Attempt at a Solution

My ans is
Ho = µ_0 = 18
H1 = µ_0 <18

Since n < 30 , and standard deviation of population unknown , so , i use t-distribution

t test = (19.1-18) / ( 2 /sqrt(2) ) = 1.905

t α = 0.05 , v =11 = 1.796

Since t test > t critical , so i reject the Ho , but accroding to the ans , i should not reject the Ho ,

Is my ans wrong ?
Your null hypothesis should be ##H_0: \mu_0 \le 18##
In addition, your calculation of the t statistic looks wrong to me.
https://en.wikipedia.org/wiki/Student%27s_t-test said:
In testing the null hypothesis that the population mean is equal to a specified value μ0, one uses the statistic where is the sample mean, s is the sample standard deviation of the sample and n is the sample size.

Homework Helper
Dearly Missed

## Homework Statement

An advertisement for a certain brand of cigarettes claimed that on average there's no more than 18mg of nicotine per cigarettes . A test of 12 cigarettes gave a sample mean of 19.1 . Assuming varience is 4 , test the claim with a significance level of α = 0.05

## The Attempt at a Solution

My ans is
Ho = µ_0 = 18
H1 = µ_0 <18

Since n < 30 , and standard deviation of population unknown , so , i use t-distribution

t test = (19.1-18) / ( 2 /sqrt(2) ) = 1.905

t α = 0.05 , v =11 = 1.796

Since t test > t critical , so i reject the Ho , but accroding to the ans , i should not reject the Ho ,

Is my ans wrong ?

You want to distinguish between values of ##\mu## that are ##\leq 18## and ##> 18##, so you should use ##H_0: \mu= 18## vs. ##H_1: \mu > 18##.

Are you sure you should use ##\sigma = 2?## If this refers to the sample variance, it is the estimated variance of ##X##, not of ##\bar{X}_{12}## = sample mean. The appropriate standard deviation for ##\bar{X}_{12}## is ##\sigma/\sqrt{12} = 2/\sqrt{12}.## That would give an even larger value of ##t## than the one you used, making ##H_0## even more strongly rejected than you indicate. This seems to be another instance of a wrong answer in the book (or possibly, the book having a weird, non-standard way of doing things).

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