Multi-Sample Data Analysis and Sampling Refinement

00:00 10 marks

A research team investigated a country’s consumer confidence index (CCI) using three independent surveys. The summary statistics for each survey are provided below:

– Survey A: sample size \(n = 50\), mean CCI = 30, standard deviation = 5.
– Survey B: sample size \(n = 100\), mean CCI = 32, standard deviation = 6.
– Survey C: sample size \(n = 75\), mean CCI = 29, standard deviation = 4.5.

Based on the data above, answer the following questions:

Part a

6 minutes 5 marks

Question

Calculate the overall weighted mean CCI from the three surveys. Discuss how the differences in standard deviations among the surveys might affect the interpretation of the combined CCI.

Hint Hint

Hint

To calculate the weighted mean, multiply each survey’s mean CCI by its sample size, sum these products, and then divide by the total sample size. Consider how a larger standard deviation indicates greater variability within a survey’s data, potentially impacting its reliability.

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Complete Answer

The weighted mean CCI is calculated as \(\frac{(50 \times 30) + (100 \times 32) + (75 \times 29)}{50 + 100 + 75} = \frac{6875}{225} \approx 30.56\). The standard deviations (5, 6, and 4.5) indicate varying levels of data dispersion. Survey B, with the highest standard deviation (6), shows the most variability, which might suggest its mean is less precisely estimated compared to Surveys A and C.

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Part b

4 minutes 5 marks

Question

To improve the reliability of inferences from combining these surveys, propose two distinct advanced sampling refinements. Explain how each refinement addresses potential limitations arising from differing sample sizes and data variability.

Hint Hint

Hint

Consider sampling methods that can enhance representation across subgroups and reduce selection bias. Think about stratified sampling, cluster sampling, or using weights based on survey reliability.

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Two advanced sampling refinements are: 1) Stratified sampling: Divide the population into subgroups based on relevant characteristics (e.g., income level, region) and sample proportionally from each stratum to ensure representation and reduce variability within strata. 2) Reliability-weighted sampling: Assign weights to each survey based on its inverse variance (or standard deviation squared). Surveys with lower variance (higher reliability) receive greater weight in the combined estimate, improving the overall inference reliability.

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