A method currently used by doctors to screen women for possible breast cancer fa

Question

A method currently used by doctors to screen women for possible breast cancer fails to detect cancer in 20% of women who actually have the disease. A new method has been developed that researchers hope will be able to detect cancer more accurately. A random sample of 80 women known to have breast cancer were screened using the new method. Of these, the new method failed to detect cancer in 9. Is the sample size sufficiently large to conduct this test of hypothesis? Explain. Be sure to state your hypotheses, denote the claim, give the critical value(s), calculate the test statistic, give your decision, and state your conclusion (in other words, a complete hypothesis test!).

Explanation / Answer

here sample size=80 and

p=proportion of fails to detect cancer=0.2

x=doctors to screen women for possible breast cancer fails to detect cancer

here we use binomial distribution with null hypothesis

null hypothesis H0:P=0.2 and

alternate hypothesis H1:P<0.2

now we use standard normal distribution z=(p-P)/sqrt(P(1-P)/n)=(0.1125-02)/sqrt(0.2*(1-0.2))=-1.96

since one tailed critical z(0.05) =1.645 is less than absolute value of calculated z=1.96, so we fail to accept H0 and conclude that new method has been developed that to detect cancer more accurately.

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