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?

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.