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For the random sample of 100 in table 1.0, the 95% confidence interval for the mean has a Lower bound of 29527.49 and an upper bound of 45209.49. The 90% confidence interval for the mean has a lower bound of 30807.15 and an upper bound of 43929.84, as shown in table 2.0. For the random sample of 400 in table 3.0, the 95% confidence interval for the mean has a Lower bound of 41657.62 and an upper bound of 49583.45. The 90% confidence interval for the mean has a lower bound of 42297.12 and an upper bound of 48943.95, as shown in table 4.0. The variable Age of respondent has a mean of 48.52, as shown in table 5.0.
The confidence interval from the above results shows that it is varying from sample to sample. The width of the confidence interval in a random sample of 100 is larger than that of a random sample of 400. This is because a larger sample makes statistics more reliable than smaller samples (Wang et al., 2018). With higher levels of the confidence interval, the width of the confidence interval is also large (Frankfort-Nachmias et al., 2020). From the above tables, the width of the confidence intervals at 95% is higher than those of 90%.
Confidence intervals are under-utilized because they need to be incorporated in day-to-day activities involving sampling and generalization. Confidence interval uses the sample size and the potential variation to make inferences of a given population by providing estimates of the range where the actual result lies (Naimi & Whitcomb, 2020). In this case, the confidence interval shows where the average income of the population lies at different confidence percentages. When confidence intervals are not used, there is a higher risk of making wrong inferences on the population because it is ordinarily impossible to enumerate all the items in the population (Tavares et al., 2020).
References
Frankfort-Nachmias, C., Leon-Guerrero, A., & Davis, G. (2020). Social statistics for a diverse society (9th ed.).
Naimi, A., & Whitcomb, B. (2020). Can Confidence Intervals Be Interpreted? American Journal of Epidemiology, 189(7), 631-633.
Tavares, D., Moura, J., Acevedo-Trejos, E., Crawford, R., Makhado, A., & Lavers, J. et al. (2020). Confidence intervals and sample size for estimating the prevalence of plastic debris in seabird nests. Environmental Pollution, 263, 114394.
Wang, X., Reich, N., & Horton, N. (2018). Enriching Students Conceptual Understanding of Confidence Intervals: An Interactive Trivia-Based Classroom Activity. The American Statistician, 73(1), 50-55. Web.
Appendix
Table 1.0: 95% confidence interval for a random sample of 100
Table 2.0: 90% confidence interval for a random sample of 100
Table 3.0: 95% confidence interval for a random sample of 400
Table 4.0: 90% confidence interval for a random sample of 400
Table 5.0: Mean age of respondent
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