# Chapter 8 - Hypothesis Testing - 8-3 Testing a Claim About a Proportion - Basic Skills and Concepts: 22

There is not sufficient evidence to support that the majority of patients are smoking one year after their treatment.

#### Work Step by Step

$H_{0}:p=50$%=0.5. $H_{a}:p>0.5.$ $\hat{p}$ is the number of objects with a specified value divided by the sample size. Hence $\hat{p}=\frac{x}{n}=\frac{39}{39+32}=0.5493.$ The test statistic is:$z=\frac{\hat{p}-p}{\sqrt{p(1-p)/n}}=\frac{0.5493-0.5}{\sqrt{0.5(1-0.5)/71}}=0.83.$ The P is the probability of the z-score being more than 0.83 which is 1 minus the probability of the z-score being less than 0.83, hence:P=1-0.7967=0.2033. If the P-value is less than $\alpha$, which is the significance level, then this means the rejection of the null hypothesis. Hence:P=0.2033 is more than $\alpha=0.05$, hence we fail to reject the null hypothesis. Hence we can say that there is not sufficient evidence to support that the majority of patients are smoking one year after their treatment.

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