STA1DCT - Birth Weight Analysis of Rat-Pups (Excel-Based Statistics)

IMPORTANT NOTE : The total possible marks for this assignment is 50. There are 40 marks associated with accuracy (i.e. correctness of your answers; the breakdown of these marks is indicated on this question sheet), a further five marks for completeness (you will only get the full five marks for completeness if you make a serious attempt to answer every question) and a further five marks for your written communication (e.g. clarity, spelling, grammar, correct use of notations etc.) STA1DCT: 40 + 5 + 5 = 50 marks.

Questions

1. Consider n sample values denoted x1, x2, . . . , xn−1, xn. Throughout this question let

denote the average of these values. For this question, you need to match the expression in the left column below with the equivalent expression in the right hand column. For convenience, the expressions on the left are labeled with numbers (e.g. Expression 1, Expression 2, . . .) and the expressions on the right with letters (e.g. Expression A, Expression B, . . .).

2. The birth weight of 800 rat-pups were recorded and are stored in the Excel file called Birth Weight.xlsx which can be downloaded from LMS. The data consists of a single column with the heading “Birth Weight”. You are required to use Excel to answer the questions below and we will treat this data as population data for this question.

(a) Calculate the population average of the birth weights provided in the Excel file. What is the population average and what was the Excel formula that you used to calculate it?

(b) Calculate the population median of the birth weights provided in the Excel file. What is the population median and what was the Excel formula that you used to calculate it?

(c) Calculate the population standard deviation of the birth weights provided in the Excel file. What is the population standard deviation and what was the Excel formula that you used to calculate it?

(d) The empirical rule in statistics states, that if the data in the population is approximately normally distributed then approximately 68% of the values in a population should fall within the interval µ ± σ (where µ and σ are the population average and population standard deviation respectively).

(i) If the birth weights of the rat-pups are approximately normally distributed then how many of the 800 values would you expect to fall within the interval µ ± σ?

(ii) You must complete this question in Excel. How many of the 800 birth weights in the Excel file fall within the interval µ ± σ? 

(iii) What percentage of the 800 birth weights fall within the interval µ ± σ?

(iv) Do you think there is evidence to suggest that the birth weights of the rat-pups are approximately normally distributed?

3. Consider again the data considered in the previous question. For this question, we will consider a random sample of 8 birth weights that have been randomly chosen from the population data provided in the Excel file. The randomly chosen data values that you should use for this question are 6.80, 6.10, 6.59, 5.07, 5.27, 6.42, 6.86, 5.32.

(a) Calculate by hand (i.e. without using Excel) the sample average. As well as providing your answer, you must also provide your workings and these must be clear and accurate. Round your final answer to 2 decimal places.

(b) Calculate by hand (i.e. without using Excel) the sample median. As well as providing your answer, you must also provide your workings and these must be clear and accurate. Round your final answer to 2 decimal places.

(c) Calculate by hand (i.e. without using Excel) the sample variance. As well as providing your answer, you must also provide your workings and these must be clear and accurate. Round your final answer to 6 decimal places.

4. Four histograms are displayed on the next page. These histograms are labelled, in order from top to bottom, Histogram A, Histogram B, Histogram C and Histogram D respectively. Refer to these histograms when answering this question.

(a) In random order, the averages of the data displayed in each of the histograms are 0, -80, -40 and 80.

(b) Similarly and again in a random order, the standard deviations of the data displayed in each of the histograms are 10, 30, 40 and 20. Your task is to study the histograms and then to match each histogram with its correct average and standard deviation. Each average and standard deviation belongs to just one histogram. When you are confident with your answers then complete the below.

5. The price of Bitcoin on the 13th of Nov 2021 was $64402.50 US dollars, and on the the 31st of Dec 2022, the price of Bitcoin was $16529.20 US dollars. Calculate, by hand, the percentage decrease in the price of Bitcoin from the 13th of Nov 2021 to the 31st of Dec 2022. You must provide detailed workings below that show how you calculated the answer. Further, your percentage answer must be rounded to 2 decimal places.

Summary of Assessment Requirements

The assessment carried a total of 50 marks, broken into:

• 40 marks for accuracy (correctness of solutions),
• 5 marks for completeness (answering all questions),
• 5 marks for communication (clarity, correctness of notation, and presentation).

Key Components of the Assessment:

1. Mathematical Expressions – Match algebraic/statistical expressions with their simplified equivalents.
2. Population Data Analysis using Excel – Calculate mean, median, and standard deviation; apply the empirical rule to rat-pup birth weight data.
3. Sample Data Analysis by Hand – Work manually with a small dataset to calculate sample mean, median, and variance (with detailed workings).
4. Histogram Interpretation – Match histograms with the correct averages and standard deviations.
5. Applied Problem – Calculate the percentage decrease in Bitcoin price over a given time period.

Academic Mentor’s Step-by-Step Guidance

The mentor adopted a systematic approach, guiding the student to build confidence in each section.

Step 1 Understanding Mathematical Expressions (Q1)

• Mentor’s Approach: Began by revisiting the definition of sample mean μ^. Broke down each expression into simple components and related them back to μ^.
• Guidance: Showed how to replace summations with nμ^ and simplify terms. Provided worked examples for Expression 1 and let the student attempt Expressions 2–5 with guidance.

Step 2 Using Excel for Population Analysis (Q2)

• Mentor’s Approach: Demonstrated the use of Excel formulas step by step.
• Guidance:
  • Taught use of =AVERAGE(range) for mean, =MEDIAN(range) for median, and =STDEV.P(range) for population standard deviation.
  • Explained the empirical rule and showed how to compute expected vs actual values within μ±σ\mu \pm \sigmaμ±σ.
  • Encouraged the student to compare theoretical percentage (68%) with actual Excel output to assess normality.

Step 3 Manual Sample Analysis (Q3)

• Mentor’s Approach: Shifted from Excel automation to manual calculations to strengthen fundamental understanding.
• Guidance:
  • Walked through formula for sample mean: xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}xˉ=n∑xi.
  • Guided sorting data to compute median.
  • Carefully explained variance formula s2=∑(xi−xˉ)2n−1s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}s2=n−1∑(xi−xˉ)2 and emphasized clear step-by-step workings.

Step 4 Histogram Interpretation (Q4)

• Mentor’s Approach: Encouraged visual thinking by analyzing shape, spread, and center of histograms.
• Guidance:
  • Matched wider spreads with higher standard deviations and centered peaks with given averages.
  • Explained reasoning behind each choice rather than just giving answers.

Step 5 Applied Percentage Change (Q5)

• First, the mentor and student noted the two Bitcoin prices:
  • On 13th Nov 2021 → 64,402.50 USD
  • On 31st Dec 2022 → 16,529.20 USD
• They worked out the difference between the two prices:
  • 64,402.50 − 16,529.20 = 47,873.30
• Next, they compared this difference to the original price by seeing what share of 64,402.50 the decrease (47,873.30) represents.
• Finally, they expressed this share as a percentage and rounded it to two decimal places, arriving at:
  • 74.33?crease

Final Outcome & Learning Objectives Covered

By the end of the session:

• Student successfully solved all parts with accuracy, ensuring completeness and clarity in written communication.

• Key Learning Objectives Achieved:

  1. Application of algebraic manipulation in statistics.
  2. Use of Excel for population data analysis (mean, median, SD).
  3. Manual computation of sample statistics (mean, median, variance).
  4. Interpretation of histograms in terms of central tendency and spread.
  5. Application of percentage change formulas in real-life scenarios.
  6. Development of academic communication skills (notation, clear workings, presentation).

Overall, the structured guidance ensured the student not only completed the assignment but also understood the logic behind each solution, strengthening both conceptual and practical statistical skills.

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