Question

It is claimed that a new treatment of a very serious disease will substantially reduce the mortality of that disease from its present rate of 25%.We observe an SRS of 100 patients who have submitted to the new treatment and note that only 10 of them die from this disease. Does the new treatment seem to make a significant improvement?

Answer 1

[Used R-Software]

R-commands and outputs:

Claim: new treatment will substantially reduce the mortality

# Present mortality rate=25% = 0.25
p0=0.25
q0=1-p0
q0
[1] 0.75
n=100 # sample size (number of patients who have submitted to new treatment)
x=10 # number of people who died
p=x/n # sample proportion
p
[1] 0.1

Hypothesis:
Null Hypothesis-H0: The hypothesis of no difference is called null hypothesis. According to Prof.R.A.Fisher, "Null hypothesis is the hypothesis which is tested for the possible rejection under the assumption that it is true".

Alternative Hypothesis-H1: The complementary hypothesis to the null hypothesis is called Alternative hypothesis.

H0: p=p0=0.25
H1: p<p0 i.e. p < 0.25

Alternative Hypothesis is one sided: Only values less than 0.25 are to be tested. That means, values in only ONE direction are to be tested. So, it is one-sided test.

Using test of single proportion:

Test statistic, z=(p-p0)/sqrt(p0*q0/n), follows Standard Normal Distribution.

z=(p-p0)/sqrt(p0*q0/n)
z
[1] -3.464102
pnorm(z)
[1] 0.0002660028

# pnorm() command in R-Software give probabilities of P(X<=x), X follows Normal Distribution.

# p-value=observed level of significance=P(Z<-3.464102)=0.0002660028

# As p-value is nearly zero, we Reject Null hypotheis. And, conclude that,p < 0.25; Yes, the new treatment seem to make a significant improvement