STA2023 Example For Application 2 Sample Data And Inference

Sta2023 Example For Application 2 Sample Data And Inferential Statis

Using the provided data and instructions, this paper aims to analyze a data set related to the weights of 5-year-old children, applying descriptive and inferential statistics techniques to interpret the data meaningfully. The goal is to utilize statistical methods, including measures of central tendency, variability, and confidence intervals, to draw inferences about the population from which the sample was drawn.

The data set consists of 25 observations, generated assuming a normally distributed population with a mean of 39.5 pounds and a standard deviation of 2.9 pounds. The data was produced via a Gaussian random number generator, providing a realistic simulation for statistical analysis. The primary focus will involve summarizing the dataset with relevant descriptive statistics, identifying outliers through range and usual value calculations, and then estimating population parameters with confidence intervals at specified confidence levels.

Introduction

Statistical analysis involving sample data allows researchers to make inferences about larger populations, especially when collecting data from the entire population is impractical or impossible. The key techniques employed include descriptive statistics, which describe the features of the data, and inferential statistics, which extend these findings to make generalized conclusions. In this context, analyzing the weights of 5-year-old children provides insights into population characteristics such as average weight, variability, and the range of typical weights.

This analysis begins by summarizing the data with measures like mean, median, standard deviation, and range. Outliers are assessed using statistical thresholds such as the usual value bounds, calculated as mean ± 2 standard deviations. Confidence intervals further quantify the plausible range for the population mean at a chosen confidence level, providing a measure of certainty around the estimate. These statistical procedures help in understanding the variability and central tendency of the data, which is essential for designing health policies and evaluating growth standards for children.

Descriptive Statistics

The primary step involves calculating descriptive statistics for the data set. The sample size (n) is 25, with the data generated around a mean (ð‘¥) of 41.72 pounds, a median of 41.9 pounds, and a standard deviation (s) of approximately 8.08 pounds. The variance (s²) is thus 65.33. The data's range, which measures the spread from the minimum to maximum values, is 31.8 pounds. These measures provide foundational insights into the data distribution.

For this sample, the mean weight (41.72 pounds) centered within the typical range indicates that most children's weights are clustered around this average. The median value (41.9 pounds) closely aligns with the mean, suggesting a symmetric distribution. The standard deviation indicates that approximately 68% of the weights fall within 8.08 pounds of the mean, reflecting moderate variability.

Assessment of Outliers and Usual Values

To identify potential outliers, the calculation of usual value bounds is performed using the formula: mean ± 2 × standard deviation. This yields a minimum usual value of 25.56 pounds and a maximum usual value of 57.88 pounds. Values outside this interval are considered atypical and possibly outliers. Using these thresholds, we find that most data points are within the typical weight range for 5-year-old children, with only extreme observations potentially flagged for further investigation.

Confidence Interval Estimation

Estimate the population mean weight using a 98% confidence interval. The confidence level of 98% implies that, if this sampling process were repeated multiple times, approximately 98% of the calculated intervals would contain the true population mean. The critical value (E) associated with this confidence level and degrees of freedom (n-1) is approximately 2.492, based on t-distribution tables.

The margin of error (E) is calculated as 2.492 multiplied by the standard error (s/√n), where the standard error (SE) is 8.08/√25 ≈ 1.616. Consequently, the margin of error E ≈ 2.492 × 1.616 ≈ 4.03 pounds.

The confidence interval is thus constructed as: mean ± E, resulting in a range from approximately 37.69 pounds to 45.75 pounds. This interval suggests that the true mean weight of 5-year-old children in the population is likely within these bounds with 98% confidence.

Interpretation of Results

The statistical analysis indicates that the average weight of 5-year-old children in the studied population is approximately 41.72 pounds, with a plausible range between 37.69 and 45.75 pounds for the true population mean. The narrow confidence interval reflects sufficient precision, given the sample size and variability. Outlier detection confirms that most weights fall within the expected typical range, affirming the normal distribution assumption.

These findings have practical implications. Health professionals and pediatricians can compare individual weights to this population standard to identify children at risk of underweight or overweight conditions. Additionally, such statistical summaries are essential when developing growth charts and assessing nutritional programs.

Conclusion

This analysis underscores the importance of descriptive and inferential statistical methods in understanding population characteristics based on sample data. The use of measures like mean, median, and standard deviation provides a snapshot of the dataset, while confidence intervals offer a probabilistic range for the population parameter. Proper interpretation of these statistics facilitates informed decision-making in pediatric health management and policy formulation.

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