My hen 'Yolko Ovo' lays from 1 to 5 eggs per day. She?s equally likely to lay 1, 2, 3, 4, or 5 eggs. At the end of every week, I count how many eggs Yolko Ovo laid during the week, and write down her weekly total. I've been doing that for many years now. If I plot a histogram of these weekly totals, which of the following would it most closely resemble?

Question

BlackTom AI · Accepted Answer

Restating the scenario: Yolko Ovo lays between 1 and 5 eggs each day with equal probability, and we sum the eggs over a 7-day week to get a weekly total. The question asks what the histogram of these weekly totals would most closely resemble.

Option analysis:
- If we were to expect a normal distribution, we should justify why the sum of several independent, identically distributed random daily counts tends to be bell-shaped. Here, each day is an independent draw from a discrete uniform distribution on {1,2,3,4,5} with mean 3 and variance 2. Over a week, the total is the sum of 7 such independent daily counts.
- The central limit theorem tells us that the sum (or average) of a reasonably large number of independent random variables with finite mean and variance tends to a normal distribution, regardless of the original distribution shape. While seven days is not extremely large, it is enough for the weekly total to start looking near-normal, especially since the daily distribution is symmetric and bounded.
- Calculating key parameters helps contextualize the shape: the weekly mean is 7 × 3 = 21, and the weekly variance is 7 × 2 = 14, giving a standard deviation of about sqrt(14) ≈ 3.74. The possible weekly totals range from 7 (all days 1) to 35 (all days 5). With these finite bounds and the additive nature of the sum, the histogram would be centered around 21 and exhibit a roughly bell-shaped dispersion around that center, rather than a flat, uniform, or highly skewed shape.

Why this reasoning supports the chosen conclusion:
- The daily counts are not normally distributed themselves, but the sum over 7 days tends toward normality due to the central limit theorem.
- The distribution will be approximately bell-shaped rather than strictly uniform or heavily skewed, given the aggregation of independent daily outcomes.

Why other potential interpretations would be incorrect (if exploring alternative options):
- It would not be uniform: a uniform weekly total would require each value to be equally likely across a wide range, which is not the case when summing independent bounded variables.
- It would not be highly skewed: the daily distribution is symmetric around 3, and summing symmetric, bounded variables tends to produce a symmetric, bell-like shape around the weekly mean.
- It would not be bimodal or exotic: with independent daily counts, the total tends to smooth out into a single peak near the mean as the number of days increases.

PSYC20007_2025_SM2 Goals of Statistical Testing II

View Explanation

Log in for full answers

Similar Questions

The basis for using a normal probability distribution to approximate the sampling distribution of and is _____.

If repeated random samples of size 40 are taken from an infinite population, the distribution of sample means

How does the normal distribution emerge? Pick between A , B and C

Can we assume that the distribution of mean lead concentrations in blood collected from repeated samples of 52 police officers to be normal?

Check the conditions required to complete this test. Specifically, is the sampling distribution of average differences approximately normally distributed?

Which of the following is incorrect about the sampling distribution of the sample mean using the Central Limit Theorem?[Fill in the blank]

More Practical Tools for International Students

Exam Browser Assistant

Stylized Writing Assistant

Plagiarism Checker Assistant

Citation Assistant

In-Class Translation Assistant

Class Notes Assistant

Quiz Search Assistant

Past Exam Questions by School

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler