The lengths of time (in years) it took a random sample of 32 former smokers to quit smoking permanently are listed. Assume the population standard deviation is 6.5 years. At $\alpha=0.05$, is there enough evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 13 years? Complete parts (a) through (e). $\begin{array}{cccccccc}9.4 & 13.8 & 10.2 & 20.6 & 18.2 & 21.1 & 16.7 & 21.8 \\ 7.1 & 20.1 & 22.1 & 9.1 & 13.9 & 21.6 & 15.6 & 13.4 \\ 15.8 & 20.1 & 12.8 & 9.3 & 9.6 & 8.4 & 12.4 & 9.1 \\ 20.9 & 20.9 & 19.5 & 7.9 & 7.6 & 21.5 & 8.3 & 20.6\end{array}$ (a) Identify the claim and state the hall hypothesis and alternative hypothesis. A. \[ \begin{array}{l} H_{0}: \mu>13 \\ H_{a}: \mu \leq 13 \text { (claim) } \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu \leq 13 \text { (claim) } \\ H_{a} \cdot \mu>13 \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu \geq 13 \text { (claim) } \\ H_{a}: \mu<13 \end{array} \] E. $\mathrm{H}_{0}: \mu>13$ (claim) $H_{a}: \mu \leq 13$ C. \[ \begin{array}{l} H_{0}: \mu=13 \text { (claim) } \\ H_{a} \cdot \mu \neq 13 \end{array} \] F. \[ \begin{array}{l} H_{0}: \mu \neq 13 \text { (claim) } \\ H_{a}: \mu=13 \end{array} \]
Solution
Step 1 ：The question is asking us to test the claim that the mean time it takes smokers to quit smoking permanently is 13 years. This is a hypothesis testing problem. The null hypothesis (H0) is the statement being tested. Usually the null hypothesis is a statement of 'no effect' or 'no difference'. In this case, the null hypothesis is that the mean time it takes smokers to quit smoking permanently is 13 years. The alternative hypothesis (Ha) is the statement we'll accept if we find the null hypothesis to be unlikely. Here, the alternative hypothesis is that the mean time it takes smokers to quit smoking permanently is not 13 years. So, the correct answer is C.
Step 2 ：Final Answer: The correct answer is C. The null hypothesis (H0) is that the mean time it takes smokers to quit smoking permanently is 13 years (\(\mu=13\)), and the alternative hypothesis (Ha) is that the mean time it takes smokers to quit smoking permanently is not 13 years (\(\mu \neq 13\)). So, the hypotheses are: \[\begin{array}{l} H_{0}: \mu=13 \text { (claim) } \\ H_{a} \cdot \mu \neq 13 \end{array}\]
