Question

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 4.3 years. at α = 0.01, 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). 10.7 22.7 17.7 14.4 19.6 21.7 11.8 9.9 18.4 10.7 9.5 9.6 13.3 22.3 15.4 21.4 8.1 14.2 12.6 14.4 11.2 16.3 8.7 19.3 20.8 7.2 18.9 22.4 7.1 9.4 8.3 7.8 (a) identify the claim and state the null hypothesis and alternative hypothesis. a. h₀: μ = 13 (claim) hₐ: μ ≠ 13 b. h₀: μ > 13 hₐ: μ ≤ 13 (claim) c. h₀: μ ≥ 13 (claim) hₐ: μ < 13 d. h₀: μ ≤ 13 (claim) hₐ: μ > 13 e. h₀: μ > 13 (claim) hₐ: μ ≤ 13 f. h₀: μ ≠ 13 (claim) hₐ: μ = 13 (b) identify the standardized test statistic. use technology. z = 1.64 (round to two decimal places as needed.) (c) find the p - value. use technology. p = (round to three decimal places as needed.)

Explanation:

Step1: Recall the formula for P - value in a two - tailed z - test

For a two - tailed z - test with test statistic \(z\), the \(P-\)value is \(P = 2\times(1 - \Phi(|z|))\), where \(\Phi(z)\) is the cumulative distribution function of the standard normal distribution.

Step2: Substitute \(z = 1.64\) into the formula

First, find \(\Phi(1.64)\). Using a standard normal table or technology (e.g., a calculator with a normalcdf function: for a standard normal distribution \(N(0,1)\), \(\text{normalcdf}(-\infty,1.64)\)). \(\Phi(1.64)\approx0.9495\)
Then, \(P=2\times(1 - 0.9495)\)

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Answer:

\(P = 0.101\)