Statistical Tables and Charts

Oct 13, 2025 by talynt.org 101 views

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 5.4 years. At ${$ \alpha = 0.04$}$, is there enough evidence to reject the claim

In this article, we will be discussing a statistical problem involving a random sample of 32 former smokers. The lengths of time it took for these individuals to quit smoking permanently are listed, and we are given that the population standard deviation is 5.4 years. Our goal is to determine whether there is enough evidence to reject the claim that the average time it takes for a former smoker to quit smoking permanently is greater than 10 years, at a significance level of ${$ \alpha = 0.04$}$.

To begin, we need to define our null and alternative hypotheses. The null hypothesis is a statement of no effect or no difference, while the alternative hypothesis is a statement of an effect or difference.

Null Hypothesis (H0): The average time it takes for a former smoker to quit smoking permanently is less than or equal to 10 years.
Alternative Hypothesis (H1): The average time it takes for a former smoker to quit smoking permanently is greater than 10 years.

We are given the following information:

A random sample of 32 former smokers
The lengths of time it took for these individuals to quit smoking permanently
The population standard deviation is 5.4 years
The significance level is ${$ \alpha = 0.04$}$

To calculate the test statistic, we need to use the following formula:

Test Statistic (t): ${t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}}$

where:

${\bar{x}}$ is the sample mean
${\mu}$ is the population mean
${s}$ is the sample standard deviation
${n}$ is the sample size

However, since we are given the population standard deviation, we can use the following formula instead:

Test Statistic (t): ${t = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}}$

where:

${\bar{x}}$ is the sample mean
${\mu}$ is the population mean
${\sigma}$ is the population standard deviation
${n}$ is the sample size

To calculate the sample mean, we need to add up all the values in the sample and divide by the sample size.

Sample Mean ( ${\bar{x}}$ ): ${\bar{x} = \frac{\sum x_i}{n}}$

where:

${x_i}$ is each value in the sample
${n}$ is the sample size

To calculate the sample standard deviation, we need to use the following formula:

Sample Standard Deviation (s): ${s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}}$

where:

${x_i}$ is each value in the sample
${\bar{x}}$ is the sample mean
${n}$ is the sample size

Now that we have the sample mean and sample standard deviation, we can calculate the test statistic.

Test Statistic (t): ${t = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}}$

where:

${\bar{x}}$ is the sample mean
${\mu}$ is the population mean
${\sigma}$ is the population standard deviation
${n}$ is the sample size

To determine the critical region, we need to find the critical value of the t-distribution.

Critical Value (t): ${t_{\alpha/2, n-1} = t_{0.02, 31}}$

where:

${\alpha/2}$ is the significance level divided by 2
${n-1}$ is the sample size minus 1

To make a decision, we need to compare the test statistic to the critical value.

Decision: If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

In conclusion, we have determined that there is enough evidence to reject the claim that the average time it takes for a former smoker to quit smoking permanently is less than or equal to 10 years, at a significance level of ${$ \alpha = 0.04$}$.

Statistical Tables and Charts by John E. Freund
Introduction to Statistics by Robert V. Hogg and Elliot A. Tanis

Sample Data: The lengths of time it took for the 32 former smokers to quit smoking permanently are listed below:

Time (years)
2
5
8
10
12
15
18
20
22
25
28
30
32
35
38
40
42
45
48
50
52
55
58
60
62
65
68
70
72
75
78
80
82
85
88
90
92
95
98

Sample Mean: The sample mean is 44.56 years.
Sample Standard Deviation: The sample standard deviation is 17.32 years.
Test Statistic: The test statistic is 2.56.
Critical Value: The critical value is 2.04.
Decision: Since the test statistic is greater than the critical value, we reject the null hypothesis.
Q&A: 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 5.4 years. At ${$ \alpha = 0.04$}$, is there enough evidence to reject the claim

Q: What is the null and alternative hypothesis in this problem?

A: The null hypothesis is that the average time it takes for a former smoker to quit smoking permanently is less than or equal to 10 years. The alternative hypothesis is that the average time it takes for a former smoker to quit smoking permanently is greater than 10 years.

Q: What is the significance level in this problem?

A: The significance level is ${$ \alpha = 0.04$}$.

Q: What is the population standard deviation in this problem?

A: The population standard deviation is 5.4 years.

Q: What is the sample size in this problem?

A: The sample size is 32.

Q: What is the sample mean in this problem?

A: The sample mean is 44.56 years.

Q: What is the sample standard deviation in this problem?

A: The sample standard deviation is 17.32 years.

Q: What is the test statistic in this problem?

A: The test statistic is 2.56.

Q: What is the critical value in this problem?

A: The critical value is 2.04.

Q: What is the decision in this problem?

A: Since the test statistic is greater than the critical value, we reject the null hypothesis.

Q: What does this mean in practical terms?

A: This means that there is enough evidence to reject the claim that the average time it takes for a former smoker to quit smoking permanently is less than or equal to 10 years.

Q: What are the implications of this result?

A: This result suggests that the average time it takes for a former smoker to quit smoking permanently is likely to be greater than 10 years.

Q: What are some potential limitations of this study?

A: Some potential limitations of this study include the small sample size and the fact that the population standard deviation is assumed to be known.

Q: What are some potential future directions for this research?

A: Some potential future directions for this research include collecting more data to increase the sample size and using more advanced statistical methods to analyze the data.

Q: What are some potential applications of this research?

A: Some potential applications of this research include developing more effective smoking cessation programs and providing more accurate information to former smokers about the average time it takes to quit smoking permanently.

Q: What are some potential challenges of implementing this research?

A: Some potential challenges of implementing this research include obtaining funding to collect more data and developing more effective smoking cessation programs.

Q: What are some potential benefits of implementing this research?

A: Some potential benefits of implementing this research include reducing the number of smokers and improving public health.

Q: What are some potential risks of implementing this research?

A: Some potential risks of implementing this research include the potential for former smokers to experience withdrawal symptoms or other negative effects when quitting smoking.