A survey showed that among 785 randomly selected subjects who completed four yea

Question

A survey showed that among 785 randomly selected subjects who completed four years of college, 144 smoke and 641 do not smoke (based on data from the American Medical Association). Suppose you want to test the claim that the rate of smoking among those with four years of college is less than the 27% rate for the general population.

a) state the random variable

b) state the parameter in words

c) state the null hypothesis

d) state the alternative hypothesis

e) identify the type 1 error that corresponds to the given hypothesis

f) identify the type 2 error that corresponds to the given hypothesis

g) Identify the consequences of a type I error in this situation

h) Identify the consequences of a type II error in this situation

i) What level of significance should you use and why

Explanation / Answer

ans=

State the null hypothesis.
H0: P = 0.27

State the alternative hypothesis.
H1: P < 0.27

Find the test statistic.
p^ = sample proportion = .1834
P= hypothesized proportion = .27
n= sample size

test statistic z = (p^-P) / sqrt[ P(1-P)/n]

Estimated p^ = 144 / 785 = 0.1834
Variance of proportion = P*(1-P)/n
= 0.27(0.73)/785 =0.0002511
S.D. of p is sqrt[0.000251] = 0.0158
z = ( 0.1834 - 0.27 ) / 0.0158 = -5.4627 --- test statistc

P-value = P( z < -5.4627) = 0.0000

Conclusion:
Since the P-value < 0.01 (significance level), reject H0 (the null hypothesis).
The rate of smoking among those with four years of college is less than the 27% rate for the general population.

Type I error = Rejecting the null hypothesis when it is true.
When the true rate is 27% , we decide that it is not.

Type II error = Not rejecting the null hypothesis when it is false.
When the true rate is not 27% , we do not reject this hypothesis

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