Using Your Textbook, LIRN-Based Research, And Your Study Not ✓ Solved

Using your textbook, LIRN-based research, and your study not

Using your textbook, LIRN-based research, and your study notes, answer the following questions that deal with confidence interval construction and explain the content in detail, including calculations. Be sure to include in-text citations and peer reviewed references in APA format in your discussion post. 1. Provide an explanation of the confidence interval and confidence level. 2. Determine in what situations you can use the z statistic to construct a confidence interval for the mean of a population. 3. Verify what test statistic is used for constructing a confidence interval on the population’s proportion. 4. Describe why and when t statistic is used in constructing a confidence interval for the mean of a population. 5. Read the case study “Pointsec Mobile Technology” on page 439 of Zikmund et al. (9th ed.) textbook (Chapter 17) and provide detailed answers to the questions asked. Make sure to provide numerical examples for questions 1-4 and include in-text citations and peer reviewed references in APA format. Additionally, prepare an APA alphanumeric outline for a research paper on the topic 'Lawful and Unlawful Immigration impact on Homeland Security (HLSC 520)'. The outline should include a title page, thesis statement, a conclusion of at least three sentences, and a reference page with a minimum of five scholarly sources. Within the outline, show the major points and research to be discussed, including both supportive and opposing research.

Paper For Above Instructions

1. Confidence interval and confidence level — explanation

A confidence interval (CI) is a range of values, derived from sample data, that is believed to contain the true population parameter with a specified probability called the confidence level (e.g., 90%, 95%, 99%) (Moore, McCabe, & Craig, 2017). The CI provides both a point estimate (the sample statistic) and a margin of error that reflects sampling variability and the chosen confidence level. The confidence level (1 − α) indicates the long-run proportion of such intervals that will contain the true parameter if many independent samples are drawn and intervals constructed the same way (Casella & Berger, 2002).

Numerical example (mean)

Suppose a manufacturer samples n = 100 units and observes a sample mean weight of x̄ = 50 g. The population standard deviation is known to be σ = 10 g. For a 95% CI, z_{0.025} = 1.96. The margin of error = z(σ/√n) = 1.96(10/√100)=1.96. Therefore CI = 50 ± 1.96 → (48.04, 51.96). This interval suggests we are 95% confident the true mean lies between 48.04 g and 51.96 g (Moore et al., 2017).

2. When to use the z statistic for a mean

The z statistic is appropriate for constructing a CI for a population mean when either (a) the population standard deviation σ is known and the sample comes from a normally distributed population, or (b) the sample size is large (typically n ≥ 30) so that the Central Limit Theorem justifies approximate normality of the sample mean and σ is known or well approximated (Casella & Berger, 2002; Zikmund, Babin, Carr, & Griffin, 2013). When σ is unknown and n is small, the t-distribution should be used instead.

Numerical example (z-applicable)

Given the example above (x̄=50, σ=10, n=100), z is appropriate because σ is known and n is large. CI computed using z gave (48.04, 51.96) (Moore et al., 2017).

3. Test statistic for a population proportion CI

Confidence intervals for a population proportion p use the z statistic based on the sampling distribution of the sample proportion p̂, provided the sample size is large enough that np̂ ≥ 5 and n(1 − p̂) ≥ 5 (Agresti & Coull, 1998). The standard error is SE = sqrt[p̂(1 − p̂)/n]. The approximate (1 − α) CI = p̂ ± z_{α/2}·SE. Agresti and Coull (1998) recommend adjusted intervals for better small-sample performance, but the classic z-interval is standard when sample size conditions hold.

Numerical example (proportion)

Suppose in a survey n = 200, the observed proportion supporting a policy is p̂ = 0.60. SE = sqrt(0.60.4/200) = sqrt(0.0012) = 0.03464. For 95% CI, margin = 1.960.03464 = 0.0679. CI = 0.60 ± 0.0679 → (0.532, 0.668). We are 95% confident the true support proportion is between 53.2% and 66.8% (Agresti & Coull, 1998).

4. Why and when the t statistic is used for a mean

The t statistic is used to construct CIs for a population mean when the population standard deviation σ is unknown and the sample comes from a normally distributed population (or is approximately normal for small samples). The t-distribution, with degrees of freedom df = n − 1, accounts for the additional uncertainty introduced by estimating σ with the sample standard deviation s; it has heavier tails than the normal distribution, producing wider intervals for small n (Casella & Berger, 2002).

Numerical example (t-based CI)

Suppose n = 20, sample mean x̄ = 100, sample standard deviation s = 15. For a 95% CI, t_{0.025,19} ≈ 2.093. Margin = 2.093(15/√20) = 2.0933.354 = 7.02. CI = 100 ± 7.02 → (92.98, 107.02). Because σ is unknown and n is small, the t-interval is appropriate (Moore et al., 2017).

5. Case study: Pointsec Mobile Technology (Zikmund et al., Chapter 17) — applied answers

Note: the textbook case asks students to evaluate survey/sampling results and make managerial recommendations regarding mobile security product adoption. Here we respond with plausible, instructive answers and calculations that illustrate application of CI concepts (Zikmund et al., 2013).

Question A — Estimate mean expected adoption time

Assume a sample of IT managers (n = 36) report a mean anticipated rollout time of x̄ = 8.2 months with s = 2.4 months. σ unknown, n = 36 (moderate size). Use t with df = 35. For 95% CI, t_{0.025,35} ≈ 2.030. Margin = 2.030(2.4/√36)=2.030(0.4)=0.812. CI = 8.2 ± 0.812 → (7.39, 9.01) months. Interpretation: with 95% confidence the true mean rollout time lies between ≈7.4 and 9.0 months (Zikmund et al., 2013).

Question B — Proportion likely to purchase

Suppose of n = 250 surveyed customers, 95 indicated they would purchase Pointsec. p̂ = 95/250 = 0.38. Check np̂ = 95 and n(1−p̂)=155 both ≥ 5. SE = sqrt(0.380.62/250)=sqrt(0.2356/250)=sqrt(0.0009424)=0.0307. For 95% CI margin =1.960.0307=0.0602. CI = 0.38 ± 0.060 → (0.320, 0.440). Thus true purchase intent likely between 32.0% and 44.0% (Agresti & Coull, 1998).

Question C — Hypothesis or managerial conclusion

If management required at least 40% purchase intent to proceed, the 95% CI (0.320, 0.440) includes 0.40; thus evidence is inconclusive — the lower bound is below threshold while the upper bound exceeds it. Management should consider additional research or pilot testing (Zikmund et al., 2013).

Question D — Recommendations

Recommend a pilot implementation focusing on segments with higher intent, further segmentation analysis, and collecting larger samples to narrow intervals. Use the CI results to plan conservative rollouts and to refine messaging that increases p̂ (Franzblau, 1997; Zikmund et al., 2013).

APA alphanumeric outline — Research paper: "Lawful and Unlawful Immigration impact on Homeland Security (HLSC 520)"

Title Page: Lawful and Unlawful Immigration Impact on Homeland Security — Student Name — Institution — Course — Date

Thesis statement: This paper examines how lawful and unlawful immigration differentially affect homeland security priorities, evaluating economic, social, and policy impacts and proposing balanced strategies that protect national security while upholding humanitarian and legal obligations (Alexseev, 2005; Bauder, 2017).

  1. INTRODUCTION
    1. Background on immigration and homeland security (Franzblau, 1997)
    2. Purpose and scope of paper
  2. MAIN IDEA 1: SECURITY THREATS AND UNLAWFUL IMMIGRATION
    1. Subtopic one: Illegal border crossings and criminal exploitation
      1. Research: threat assessments (Franzblau, 1997)
    2. Subtopic two: Radicalization concerns and intelligence challenges
      1. Research: case studies and policy analyses (Alexseev, 2005)
  3. MAIN IDEA 2: ROLE OF LAWFUL IMMIGRATION
    1. Subtopic one: Economic and societal contributions (Bauder, 2017)
    2. Subtopic two: Legal pathways reducing illicit flows (Cox & Rodriguez, 2015)
    3. Subtopic three: Policy trade-offs
      1. Research: reforms and outcomes (Farina, 2018)
  4. MAIN IDEA 3: POLICY RESPONSES AND RECOMMENDATIONS
    1. Subtopic one: Border security measures and civil liberties (Heimburger, 2017)
    2. Subtopic two: Community resilience and mental health impacts (McLeigh, 2010)
    3. Subtopic three: Balanced policy proposals and evaluation metrics
  5. CONCLUSION
    1. Summary of key findings
    2. Policy recommendations for integrated security and humanitarian approaches
    3. Future research directions

Conclusion (three sentences): Lawful immigration provides substantial economic and social benefits that can enhance homeland security by reducing incentives for illicit migration, while unlawful immigration presents discrete risks that require targeted, intelligence-driven responses. Effective homeland security policy should combine robust border management with legal pathways, community engagement, and protections for vulnerable populations. Future research should evaluate policy pilots and their measurable effects on security metrics and community well-being (Alexseev, 2005; Bauder, 2017; Franzblau, 1997).

References

  • Agresti, A., & Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of a binomial proportion. The American Statistician, 52(2), 119–126.
  • Alexseev, M. (2005). Immigration phobia and the security dilemma: Russia, Europe, and the United States. Cambridge University Press.
  • Bauder, H. (2017). Immigration dialectic: Imagining community, economy, and nation. University of Toronto Press.
  • Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Duxbury.
  • Cox, A., & Rodriguez, C. (2015). The president and immigration law redux. Yale Law Journal, 125(1).
  • Farina, M. (2018). White nativism, ethnic identity, and US immigration policy reforms. Routledge.
  • Franzblau, K. J. (1997). Immigration's impact on U.S. national security and foreign policy. Homeland Security Digital Library.
  • Heimburger, R. (2017). God and the illegal alien: United States immigration law and a theology of politics. Cambridge University Press.
  • McLeigh, J. (2010). How do Immigration and Customs Enforcement (ICE) practices affect the mental health of children? American Journal of Orthopsychiatry, 80(1), 96–100.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the practice of statistics (9th ed.). W.H. Freeman.
  • Zikmund, W. G., Babin, B. J., Carr, J. C., & Griffin, M. (2013). Business research methods (9th ed.). Cengage.

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