Business Statistics Assignment Week 3 Question 1 If You Coul

Business Statisticsassignment Week 3question 1if You Could Stop Time

Evaluate the following assignment questions based on the given scenario and statistical concepts:

Question 1: In a survey reported in a USA Today Snapshot, individuals' ideal ages were recorded based on their age groups, with an average ideal age of 41 for all adults. Younger than 30-year-olds tend to desire to be older, while older than 30-year-olds prefer to be younger. The age of the respondent is not the random variable; instead, the variable “ideal age” is used relative to age groups. Explain why the person's actual age is not the random variable in this context and describe how “age” is used with regard to the age group. Identify the random variable involved and discuss whether it is discrete or continuous.

Question 2: Calculate the area to the left of the following Z-values under the standard normal curve:

• a. Z = -1.30
• b. Z = -3.20
• c. Z = -2.56
• d. Z = -0.64

Paper For Above instruction

The survey conducted as part of a USA Today Snapshot extensively examines individuals' perceptions of their ideal age should they possess the ability to stop time and live eternally in good health. The data reveal intriguing trends about age-related desires, emphasizing the relationship between actual age and preferred ideal age. In analyzing such data, it is essential to identify the variables involved, their nature, and their roles within the study's statistical framework. Additionally, understanding the properties of the standard normal distribution is crucial for interpreting Z-scores and associated probabilities.

Analysis of Question 1

In the given scenario, the person’s actual age is explicitly distinguished from the variable of interest—“ideal age.” The question asks respondents what age they would choose to be if they could live forever in good health, which reflects their subjective preference rather than their objective age. Therefore, the actual age of the respondent is not the random variable because it is a fixed attribute of the individual, known without uncertainty, in this context. Instead, “ideal age” functions as the variable of interest because it varies across the population based on respondents' perceptions and desires.

The concept of “age” as used in this analysis is tied to the respondent’s subjective evaluation rather than an inherent, measurable property like chronological age. The survey stratifies participants into age groups (such as younger than 30, between 30 and 50, and older than 50) to analyze how ideal age preferences differ across age cohorts. These groupings serve to organize data but do not alter the statistical nature of the variable—“ideal age”—which remains a continuous variable that can take on any value within a range, reflecting the respondents' preferences. The categorization into age groups is primarily for descriptive and comparative purposes rather than converting the variable into a categorical or discrete measurement.

Role and Nature of the Random Variable

The random variable in this scenario is the “ideal age” chosen by each individual respondent. This variable is random because it is subject to variation across the population and can be modeled statistically to analyze its distribution. Its role is central: it encapsulates respondents' preferences and allows analysts to summarize these preferences through measures such as the mean (which is 41 years in this case), variance, and other descriptive statistics. By studying the distribution of “ideal age,” researchers can identify trends, differences among groups, and correlations between actual age and ideal age preferences.

Furthermore, the “ideal age” variable is continuous because it can theoretically take any value within a range (such as 0 to 120 years), depending on respondents' answers. Although survey data often report averages or group means, the underlying individual preferences are continuous in nature. This understanding is important for selecting appropriate statistical techniques, such as those involving the normal distribution or other continuous probability models.

Analysis of Question 2

For the second question, the focus is on understanding the areas under the standard normal curve to the left of specified Z-values. These areas correspond to probabilities in the standard normal distribution, which is symmetric and centered at zero with a standard deviation of 1. The areas represent the probability that a standard normal variable Z is less than the given Z-score.

Using standard normal distribution tables or statistical software, the following areas are obtained:

• a. Z = -1.30: P(Z
• b. Z = -3.20: P(Z
• c. Z = -2.56: P(Z
• d. Z = -0.64: P(Z

These probabilities are critical in hypothesis testing and confidence interval calculations, providing insights into how extreme a Z-score is within a normal distribution. The negative Z-values indicate corresponding areas under the curve to the left of these points, representing relatively low probabilities as the Z-scores become more negative, indicating values farther in the left tail of the distribution.

Conclusion

This analysis underscores the importance of understanding the distinction between objective data (such as actual age) and subjective variables (like ideal age), as well as recognizing the properties of the normal distribution when interpreting Z-scores. Proper identification of the random variables and their nature facilitates accurate statistical analysis and meaningful interpretation of survey results.

References

• Wasserstein, R. L., & Lazar, N. A. (2016). The ASA's Statement on p-Values: Context, Process, and Purpose. The American Statistician, 70(2), 129-133.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Brooks Cole.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman.
• Ott, L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
• Agresti, A., & Franklin, C. (2017). Statistics: The Art and Science of Learning from Data (4th ed.). Pearson.
• Ross, S. M. (2014). Introduction to Probability and Statistics (5th ed.). Academic Press.
• Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
• Schwab, C. (2018). Understanding Standard Normal Distribution and Z-Scores. Statistics How To. Retrieved from https://www.statisticshowto.com
• Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
• Johnson, N. L., & Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions. Houghton Mifflin.