Math 221 Statistics For Decision Making Week 6 Ilab Name

4math 221 Statistics For Decision Makingweek 6 Ilabname

Identify the assignment question/prompt and clean it: remove any rubric, grading criteria, point allocations, meta-instructions to the student or writer, due dates, and any lines that are just telling someone how to complete or submit the assignment. Also remove obviously repetitive or duplicated lines or sentences so that the cleaned instructions are concise and non-redundant. Only keep the core assignment question and any truly essential context.

The cleaned assignment prompt is:

Calculate and interpret confidence intervals for sleep and height data, compare intervals across different confidence levels and groups, and analyze the distribution of a variable called DRIVE, including predicted and actual percentages within certain ranges.

Paper For Above instruction

The objective of this assignment is to apply statistical concepts such as data simulation, confidence intervals, and normal probabilities to real dataset variables. Specifically, students are tasked with calculating and interpreting confidence intervals for the sleep variable, comparing these intervals at 95% and 99% confidence levels, and analyzing the height variable segmented by gender. Additionally, the assignment involves estimating the distribution of the DRIVE variable and comparing predicted probabilities with actual data.

Introduction

Statistical analysis is essential for decision-making and understanding data distributions within diverse variables. In this paper, we focus on three primary areas: confidence intervals for sleep and height, and probability predictions for the DRIVE variable. These analyses will illustrate the application of fundamental statistical methods, such as calculating means, standard deviations, and confidence intervals, as well as interpreting the results in context.

Confidence Intervals for Sleep Data

To estimate the average hours of sleep among students, the dataset's mean and standard deviation are calculated. Using Excel functions, the mean is obtained by typing =AVERAGE(E2:E36), and the standard deviation by =STDEV(E2:E36). The maximum error at a 95% confidence level is computed using the =CONFIDENCE.NORM(0.05,E38,35) function, where 0.05 corresponds to the significance level (alpha), E38 to the standard deviation, and 35 to the sample size minus one.

Once the maximum error (E) is obtained, the confidence interval is calculated as (x – E, x + E), providing an estimate within which the true population mean is likely to fall with 95% confidence. The same process is followed for a 99% confidence level, replacing 0.05 with 0.01 in the confidence function. Interpreting these intervals, the 95% CI suggests that we are 95% confident that the true mean sleep hours lie within the specified range, whereas the 99% CI, being wider, indicates a higher confidence but less precision.

The wider interval at 99% confidence occurs because increasing the confidence level increases the margin of error, reflecting greater uncertainty about the estimate.

Confidence Intervals for Height by Gender

Using pivot table data from the previous week's analysis, the mean and standard deviation of height are noted separately for males and females. For each group, the maximum error at 95% confidence is calculated using the =CONFIDENCE.T(0.05, stdev, n) formula, where stdev is the group's standard deviation and n is the group size.

The confidence intervals are derived by adding and subtracting these maximum errors from the respective group means. The comparison reveals that the interval for the group with greater variability is wider, which reflects higher uncertainty in that group's height estimation. At 99% confidence, the intervals become wider due to the increased confidence level, following similar logic as above. The group differences in interval widths are attributable to sample variability and size, impacting the precision of estimates.

Analysis of the DRIVE Variable

For the DRIVE variable, the mean and standard deviation are calculated using =AVERAGE(A2:A36) and =STDEV(A2:A36). Assuming normal distribution, the probability of a value being less than 40 miles is predicted using =NORM.DIST(40, mean, stdev, TRUE). This probability is converted to a percentage and compared with the actual percentage of data points in the dataset that are below 40 miles, identified by sorting and counting. The comparison highlights the accuracy of the normal model in estimating real data.

Similarly, the percentages of data within the ranges of 40 to 70 miles and above 70 miles are estimated using the cumulative distribution function outputs at the boundaries. The predicted percentages are contrasted with actual data obtained through counting. Differences between predicted and actual proportions can be explained by deviations from normality or sampling variability, informing on the model's suitability.

Conclusion

This application of statistical methods demonstrates how confidence intervals provide valuable estimates of population parameters with quantifiable certainty. Comparing intervals at different confidence levels emphasizes the trade-off between precision and confidence. Analyzing the DRIVE variable illustrates the practical use of normal probability distributions in predicting data proportions and assessing model accuracy. Such analyses are fundamental tools in decision-making, statistical inference, and data-driven insights across various fields.

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