Study Conducted By The Research Department Of A Pharma

For A Study Conducted By The Research Department Of A Pharmaceutical C

For a study conducted by the research department of a pharmaceutical company, 245 randomly selected individuals were asked to report the amount of money they spend annually on prescription allergy relief medication. The sample mean was found to be $17.60 with a standard deviation of $5.30 . A random sample of 210 individuals was selected independently of the first sample. These individuals reported their annual spending on non-prescription allergy relief medication. The mean of the second sample was found to be $18.40 with a standard deviation of $4.40. As the sample sizes were quite large, it was assumed that the respective population standard deviations of the spending for prescription and non-prescription allergy relief medication could be estimated as the respective sample standard deviation values given above. Construct a 95% confidence interval for the difference between the mean spending on prescription allergy relief medication () and the mean spending on non-prescription allergy relief medication (). Then complete the table below. What is the lower limit of the 95% confidence interval? What is the upper limit of the 95% confidence interval? Carry your intermediate computations to at least three decimal places. Round your answers to at least two decimal places.

Paper For Above instruction

The objective of this study is to compare the average annual spending on prescription versus non-prescription allergy relief medications among individuals. Using a large sample size from each group, the research seeks to estimate the difference between these two population means with a high level of confidence (95%). To achieve this, the construction of a confidence interval for the difference between the two means provides a statistical range in which the true difference likely resides, considering sample data and variability.

Given data include a sample of 245 individuals reporting their annual expenditure on prescription allergy medication, with a sample mean of $17.60 and a sample standard deviation of $5.30. For non-prescription allergy medications, a separate sample of 210 individuals reported an average expenditure of $18.40, with a standard deviation of $4.40. Since the sample sizes are large, it is reasonable to estimate the population standard deviations using the sample standard deviations, which allows for the application of the two-sample z-interval for the difference of means (Zar, 2010).

Calculations of the Confidence Interval

The formula for the confidence interval of the difference between two independent means when population standard deviations are estimated by sample standard deviations is:

CI = (x̄₁ - x̄₂) ± Zα/2 * √(s₁²/n₁ + s₂²/n₂)

Where:

  • x̄₁ = 17.60 is the mean expenditure for prescription medication.
  • s₁ = 5.30 is the standard deviation for prescription medication.
  • n₁ = 245 is the sample size for prescription medication.
  • x̄₂ = 18.40 is the mean expenditure for non-prescription medication.
  • s₂ = 4.40 is the standard deviation for non-prescription medication.
  • n₂ = 210 is the sample size for non-prescription medication.
  • Zα/2 = 1.96 for a 95% confidence level (from standard normal distribution table).

Computing the Standard Error (SE)

The standard error for the difference is:

SE = √(s₁²/n₁ + s₂²/n₂) = √((5.30)²/245 + (4.40)²/210) = √(28.09/245 + 19.36/210)

Calculating the components:

  • 28.09/245 ≈ 0.1147
  • 19.36/210 ≈ 0.0922

Sum: 0.1147 + 0.0922 ≈ 0.2069

SE ≈ √0.2069 ≈ 0.4546

Determining the Margin of Error (ME)

ME = Zα/2 SE = 1.96 0.4546 ≈ 0.8910

Calculating the Confidence Interval

The difference in sample means:

x̄₁ - x̄₂ = 17.60 - 18.40 = -0.80

The confidence interval is then:

Lower limit: -0.80 - 0.8910 ≈ -1.69

Upper limit: -0.80 + 0.8910 ≈ 0.09

Therefore, the 95% confidence interval for the difference in mean spending is approximately from -1.69 to 0.09 dollars.

The lower limit of the interval is -1.69, and the upper limit is 0.09.

Implication

This interval suggests that there is no statistically significant difference in average spending on prescription versus non-prescription allergy medications at the 95% confidence level, as the interval includes zero. This implies that the true difference could be zero or very close to it, indicating similar spending habits on these two types of medications among the sampled population.

References

  • Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson.
  • Dean, A. G., et al. (2013). OpenEpi: Open Source Epidemiologic Statistics for Public Health, Version can be found at www.OpenEpi.com.
  • Franklin, C. (2014). Confidence intervals: concepts and calculations. Journal of Statistical Computation and Simulation, 84(2), 347–359.
  • Higgins, J. P. T., & Green, S. (Eds.). (2011). Cochrane Handbook for Systematic Reviews of Interventions. The Cochrane Collaboration.
  • LaMorte, W. W. (2017). Confidence intervals. Boston University School of Public Health. Retrieved from https://sphweb.bumc.bu.edu/.
  • Newcombe, R. G. (1998). Two-sided confidence intervals for the difference between independent proportions: Comparison of seven methods. Statistics in Medicine, 17(8), 873–890.
  • Cohen, J. (1990). Things I Have Learned (So Far). American Psychologist, 45(12), 1304–1312.
  • Johnson, R. A., & Wichern, D. W. (2014). Applied Multivariate Statistical Analysis (6th ed.). Pearson.
  • Moore, D. S., & McCabe, G. P. (2006). Introduction to the Practice of Statistics (5th ed.). W. H. Freeman.
  • Woodward, M. (2013). Epidemiology: Study Design and Data Analysis. CRC Press.

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