Questions: Question 8, 7.2.35-T Part 2 of 4 The lengths of time (in years) it took a random sample of 32 former smokers to quit smoking permanently are listed. Assume the evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 13 years? Complete parts (a) thr 13.5 8.4 19.3 12.8 15.5 20.5 13.6 7.6 11.6 20.9 18.7 17.7 12.5 11.4 16.3 19.5 11.7 16.4 10.7 8.3 16.4 10.3 10.1 13.9 13.9 8.4 12.7 21.8 19.7 19.3 13.8 7.5 (a) Identify the claim and state the null hypothesis and alternative hypothesis. A. H0: μ ≤ 13 (claim) Ha: μ > 13 D. H0: μ > 13 Ha: μ ≤ 13 (claim) E. H0: μ > 13 (claim) Ha: μ ≤ 13 b) Identify the standardized test statistic. Use technology. (Round to two decimal places as needed.)

Question 8, 7.2.35-T Part 2 of 4

The lengths of time (in years) it took a random sample of 32 former smokers to quit smoking permanently are listed. Assume the evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 13 years? Complete parts (a) thr

13.5 8.4 19.3 12.8 15.5 20.5 13.6 7.6
11.6 20.9 18.7 17.7 12.5 11.4 16.3 19.5
11.7 16.4 10.7 8.3 16.4 10.3 10.1 13.9
13.9 8.4 12.7 21.8 19.7 19.3 13.8 7.5

(a) Identify the claim and state the null hypothesis and alternative hypothesis.

A. H0: μ ≤ 13 (claim)
Ha: μ > 13

D. H0: μ > 13 Ha: μ ≤ 13 (claim) E. H0: μ > 13 (claim) Ha: μ ≤ 13

b) Identify the standardized test statistic. Use technology.
(Round to two decimal places as needed.)

Transcript text: Question 8, 7.2.35-T Part 2 of 4 The lengths of time (in years) it took a random sample of 32 former smokers to quit smoking permanently are listed. Assume the evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 13 years? Complete parts (a) thr \begin{tabular}{cccccccc} 13.5 & 8.4 & 19.3 & 12.8 & 15.5 & 20.5 & 13.6 & 7.6 \\ 11.6 & 20.9 & 18.7 & 17.7 & 12.5 & 11.4 & 16.3 & 19.5 \\ 11.7 & 16.4 & 10.7 & 8.3 & 16.4 & 10.3 & 10.1 & 13.9 \\ 13.9 & 8.4 & 12.7 & 21.8 & 19.7 & 19.3 & 13.8 & 7.5 \end{tabular} (a) Identify the claim and state the null hypothesis and alternative hypothesis. A. $\mathrm{H}_{0}: \mu \leq 13$ (claim) \[ H_{a}: \mu>13 \] D. $H_{0}: \mu>13$ $H_{a}: \mu \leq 13$ (claim) E. $H_{0}: \mu>13$ (claim) $H_{a}: \mu \leq 13$ b) Identify the standardized test statistic. Use technology. $\square$ (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Sample Mean

The sample mean $ \bar{x} $ is calculated as follows: \[ \bar{x} = \frac{\sum_{i=1}^N x_i}{N} = \frac{454.7}{32} = 14.21 \]

Step 2: Calculate the Sample Variance and Standard Deviation

The sample variance $ s^2 $ is given by: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} = 18.22 \] The sample standard deviation $ s $ is: \[ s = \sqrt{18.22} = 4.27 \]

Step 3: Calculate the Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} = \frac{4.27}{\sqrt{32}} = 0.7548 \]

Step 4: Calculate the Test Statistic

The test statistic $ Z_{\text{test}} $ for the hypothesis test is calculated as: \[ Z_{\text{test}} = \frac{\bar{x} - \mu_0}{SE} = \frac{14.21 - 13}{0.7548} = 1.603 \]

Step 5: Calculate the P-value

For a right-tailed test, the P-value is calculated as: \[ P = 1 - T(z) = 0.0545 \]

Final Answer

The null hypothesis $ H_0: \mu \leq 13 $ is tested against the alternative hypothesis $ H_a: \mu > 13 $. The test statistic is $ Z_{\text{test}} = 1.603 $ and the P-value is $ 0.0545 $.

Since the P-value $ 0.0545 $ is slightly greater than the significance level $ \alpha = 0.05 $, we do not reject the null hypothesis.

Thus, the final answer is: \[ \boxed{Z_{\text{test}} = 1.603, \, P = 0.0545} \]

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