Prospective Jurors For Court Trials Are Selected From A Jury

1 Prospective Jurors For Court Trials Are Selected From a Jury Pool

Prospective jurors for court trials are selected from a jury pool summoned each day. The pool is drawn from the population of registered voters in the county. For each summon issued and also for each selection of a juror from the jury pool, let ï° denote the probability that a person living in the east side of the county is selected. a) A random sampling of 12 prospective jurors contains 1 person from the east side. Report the P-value for testing the null hypothesis that ï°=0.2 using Wald test. This null hypothesis is suggested by that the fact that 20% of the population of the registered voters in this county living in the east side. b) The entire jury pool consisting of all 400 registered voters who were summoned to report for possible jury duty, there were 68 persons from the east side. Construct a 90% Wald confidence interval for ï°. The y-intercept of a regression equation is interpreted as: A) The predicted value of y when x equals 0. B) The change in y for a one unit change in x. C) The amount of variation in y explained by x. Which of the following correlations would have the weakest linear relationship? A) r = 0.9 B) r = -0.7 C) r = -0.1 D) r = 0.6 In general, which is more likely to contain the unknown population mean? A) A 90% confidence interval B) A 95% confidence interval C) A 99% confidence interval D) They are all equally likely Suppose a 95% confidence interval for the proportion of Americans who smoke is 0.178 to 0.242. Which one of the following statements is NOT true? A) It is reasonable to say that more than 25% of Americans smoke. B) It is reasonable to say that less than 25% of Americans smoke. C) It is reasonable to say that more than 17% of Americans smoke. D) An "acceptable" hypothesis is that about 19% of Americans smoke. Suppose there is a project to study the effectiveness of a new treatment for high cholesterol. To test whether the new treatment lowers cholesterol level, cholesterol readings were recorded on same participant before and after a 10-week treatment period. What kind of test would you recommend for this study? A) a two sample t-test B) a paired t-test C) a two proportion test D) an unpooled t-test 21. Identify whether the comparison is based on two independent samples or paired data: We test if the average number of hours studied for college freshman is different from the average number of hours studied for seniors in college. A) Paired B) Independent Based on the number and types of variables present, select the most appropriate display for each of the following: Rent charged (in dollars) and apartment size (in sq. ft.) of a sample of one-bedroom apartments in State College. A) Bar Graph B) Histogram C) Two-way table D) Scatterplot E) Side-by-Side Boxplots Based on the number and types of variables present, select the most appropriate display for each of the following: Whether a person is a vegetarian and Gender. A) Bar Graph B) Histogram C) Two-way table D) Scatterplot E) Side-by-Side Boxplots

Paper For Above instruction

The assignment encompasses a multifaceted analysis involving hypothesis testing, confidence interval construction, correlation interpretation, comparative testing, and data visualization. Each segment illustrates fundamental statistical concepts and their application to real-world scenarios, emphasizing understanding and proper interpretation of statistical measures.

Hypothesis Testing with Jury Pool Data

In the first scenario, we examine whether the probability ï° that a prospective juror from the east side of the county is selected aligns with the hypothesized probability of 0.2. Given that in a sample of 12 prospective jurors, only 1 is from the east side, a Wald test is appropriate for evaluating this hypothesis. The Wald test assesses whether the observed proportion significantly deviates from the hypothesized proportion by calculating a test statistic and corresponding P-value. For the sample, the observed proportion (p̂) is 1/12 ≈ 0.0833. The standard error (SE) under null hypothesis ï°=0.2 is calculated as √[0.2(1−0.2)/12], which equals approximately 0.116. The Wald test statistic (Z) is (p̂ − 0.2)/SE ≈ (0.0833−0.2)/0.116 ≈ −1.009. The P-value, derived from the standard normal distribution for this Z-value, is approximately 0.313, indicating insufficient evidence to reject the null hypothesis at typical significance levels.

Next, considering the entire jury pool of 400 registered voters with 68 from the east side, the sample proportion is 68/400 = 0.17. A 90% Wald confidence interval for ï° is constructed using the formula p̂ ± z_α/2 SE, where z_α/2 ≈ 1.645. The standard error is √[0.17(1−0.17)/400] ≈ 0.019. The confidence interval is thus 0.17 ± 1.6450.019, which ranges approximately from 0.137 to 0.203. This interval suggests that, with 90% confidence, the true proportion of east side residents in the jury pool lies within this range.

Interpretation of Regression Intercept and Correlation Strength

The y-intercept of a regression equation typically refers to the predicted value of the dependent variable when the independent variable is zero. Therefore, the correct interpretation among the options provided is A) The predicted value of y when x equals 0. The correlations listed demonstrate varying degrees of linear relationship strength. The correlation coefficient r = -0.1 indicates the weakest linear relationship among the options, as it is closest to zero, signifying minimal linear association.

Confidence Intervals and Hypotheses

In confidence interval analysis, increasing the confidence level (from 90% to 99%) generally results in a wider interval, thus increasing the likelihood of containing the true population parameter. Therefore, a 99% confidence interval is more likely to contain the true mean than a 95% or 90% interval. Among the statements regarding a 95% confidence interval for smoking prevalence, statement A) suggesting that more than 25% of Americans smoke is NOT true because the interval (0.178 to 0.242) does not include values greater than 0.25, and thus, we cannot reasonably assert that more than 25% smoke based solely on this interval.

Selecting Appropriate Statistical Tests

For assessing whether a new treatment reduces cholesterol levels measured on the same participants before and after the intervention, the appropriate statistical tool is a paired t-test. This test accounts for the dependence between pre- and post-treatment measurements within the same subjects. Conversely, a two-sample t-test would be unsuitable here because the data are paired, not independent.

Data Comparison Types and Visualization

When comparing the average hours studied by college freshmen and seniors, the data are from two independent groups; thus, the correct choice is B) Independent. When visualizing rent charged versus apartment size, a scatterplot is ideal for exploring the relationship between two quantitative variables. It reveals trends, correlations, and potential outliers. For categorical variables like vegetarian status and gender, a two-way table provides an effective cross-tabulation, aiding in understanding the association between these variables.

Conclusion

This comprehensive review demonstrates fundamental statistical methodologies, from hypothesis testing and confidence interval estimation to correlation analysis and data visualization. Mastery of these concepts allows researchers to draw meaningful inferences from data, informing decision-making in various contexts such as jury selection, medical studies, and social sciences.

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