Question

A medical researcher says that less than `25%` of US. adults are smokers. In a random sample of `200` US. adults, `18.5%` say that they are smokers. At `alpha = 0.05`, is there enough evidence to reject the researchers claim?

Answer 1

The medical researcher's expectation is wrong. There is enough statistical evidence to prove that 25% or more are smokers in the university.

Explanation:

It is about testing the significance of sample Proportions.

Population Parameter

P = 25% [Population Proportion}

Sample Statistic
p=18.5% [sample Proportions]

Significance level 0.05

Sample size `n=200`

Distribution to be used : Normal Distribution [since `n>30`]

`H_O:P>=25%`
`H_1:P<25%`

[Tips : We are going to compare Calculated `z` Value with Table `z`value]

Calculations:

Table Value (`z`) depends on the level of significance and the type of test (we are going to conduct left tail test)

In our case table value is `=1.645` [It is taken form the table]

Since left tail test is used the actual Table value is `-1.645`

Refer the image.

Calculated `z=(p-P)/(SE)`

Here `SE = sqrt(nPQ)` --------- {Were `Q = 1-P` and SE is Standard Error]

`SE=sqrt(200 xx 0.25 xx 0.75)=6.124`

`z=(18.5-25)/6.124=(-6.5)/6.124=-1.06`

`-1.06`is the calculated `z` value

Conclusion:

Since the calculated value `(-1.06)` is greater than the table value `(-1.645)` , Null Hypothesis is accepted.

Inference:

There is enough statistical evidence to prove that 25% or more are smokers in the university.

It means, the medical researcher's expectation is wrong.