The population distribution from which the samples were drawn is symmetric and mound-shaped with the top of the mound at 32. 95% of the data is between 30 and 34, and 99.796% of the data is between 29 and 35. How well does each histogram reflect these characteristics? First, consider if the histograms are symmetric and mound-shaped, with the top of the mound at 32. Recall that a mound-shaped distribution has a peak or mound near the middle of the data, and a symmetric distribution means both sides are similar when the graph is folded down the middle: Sample of size 30 Sample of size 50 Sample of size 100 The three histograms show a mound-shaped distribution, and the data gradually reduces on either side of the mound in a select fashion. The highest bar signifies the top of the mound. In histogram (i), the highest bar occurs at a value of x of 32. In histogram (ii), the highest bar occurs at a value of x of units. In histogram (iii), the highest bar occurs at a value of x of units
The Correct Answer and Explanation is:
Let’s walk through the problem carefully and then provide a thorough explanation of how each histogram reflects the described distribution characteristics.
Correct Answers for the Blanks in the Image:
- The three histograms show a mound-shaped distribution, and the data gradually reduces on either side of the mound in a symmetric fashion.
- In histogram (i), the highest bar occurs at a value of x of 32.
- In histogram (ii), the highest bar occurs at a value of x of 32 units.
- In histogram (iii), the highest bar occurs at a value of x of 32 units.
Thorough 1000-Word Explanation:
Understanding the Distribution:
The question is based on a classic symmetric and mound-shaped distribution (a normal distribution). The known characteristics are:
- The distribution is symmetric and centered at 32.
- 95% of the data lies between 30 and 34.
- 99.7% of the data lies between 29 and 35.
This is consistent with the Empirical Rule (or 68–95–99.7 rule) in statistics, which states:
- 68% of the data falls within 1 standard deviation from the mean.
- 95% within 2 standard deviations.
- 99.7% within 3 standard deviations.
This implies that:
- The mean (μ) is 32.
- One standard deviation (σ) is likely 1, because:
- 95% of data falls between 30 and 34 → this is ±2 units from the mean → 2σ = 2 → σ = 1.
So, we’re evaluating how well the sample histograms mimic a population that is normally distributed with μ = 32, σ = 1.
Histogram Evaluation:
We now evaluate each histogram in terms of:
- Symmetry
- Mound-shaped nature
- Central tendency (peak at x = 32)
- Spread (data concentration within expected intervals)
Histogram (i): Sample Size = 30
- Symmetry: Somewhat symmetric but shows variability due to the small sample size.
- Left side of the mound (around 30–31) is slightly lower than the right side (33–34).
- There is a bar at 30 and 34, which supports the 95% coverage between 30 and 34.
- Mound-shaped: Yes, there’s a peak at 32 with frequencies tapering off on either side.
- Central Peak: Highest bar is at 32, matching population peak.
- Spread: Most data between 30 and 34, with a few data points at 29 and 35, aligning with the empirical rule.
Summary: Reasonably good approximation despite sample variability; less smooth due to the small sample.
Histogram (ii): Sample Size = 50
- Symmetry: Stronger symmetry than histogram (i); left and right sides are more balanced.
- Peak near 32 is flanked by bars that decrease in frequency as you move away.
- Mound-shaped: Clearly mound-shaped.
- Central Peak: Highest bar occurs at 32, again matching the population mean.
- Spread: Bars stretch from 29 to 35, perfectly in line with 99.7% data rule.
- Densest bars fall within 30–34, in accordance with the 95% rule.
Summary: This histogram reflects the characteristics very well — better than histogram (i) due to larger sample size leading to smoother representation.
Histogram (iii): Sample Size = 100
- Symmetry: Excellent symmetry. Almost perfect mirror around x = 32.
- Mound-shaped: Very smooth mound; classic bell curve representation.
- Central Peak: Highest bar is at 32, directly supporting the population’s peak.
- Spread: Bars extend from 29 to 35 with most of the data between 30 and 34.
- Perfectly matches 95% and 99.7% intervals of a normal distribution.
Summary: This histogram best reflects the true nature of the population distribution. Larger sample sizes produce smoother, more accurate approximations of population distributions.
Overall Observations:
- Symmetry Improves with Sample Size:
- Histogram (i) shows more irregularities because random variation has more impact in small samples.
- As sample size increases, the histogram becomes smoother and more symmetric due to the Law of Large Numbers, which states that the sample distribution approaches the population distribution as the sample size grows.
- Mound Shape Emerges with Larger Samples:
- All three histograms are roughly mound-shaped, but the shape becomes more obvious and textbook-like with a sample size of 100.
- Peak at 32 is Consistently Captured:
- Each histogram correctly identifies the central peak at x = 32, which reflects the given population mean.
- This suggests that even small samples can provide a good sense of central tendency.
- Spread Matches Population Expectations:
- All histograms show that data mostly lies between 30 and 34, and very few observations fall outside 29 or 35.
- This validates the stated 95% and 99.7% intervals.
Conclusion:
Each histogram approximates the population distribution to varying degrees based on sample size:
- Histogram (i) provides a basic, less smooth estimate. The peak is correct, but the symmetry and spread are affected by sample variability.
- Histogram (ii) shows a much better approximation — symmetry, peak, and spread all align well with the population characteristics.
- Histogram (iii) gives the best representation, being smooth, symmetric, and centered with a spread that perfectly aligns with the empirical rule.
Therefore, larger sample sizes lead to better approximations of the population distribution, and all three histograms generally support the claim that the underlying distribution is symmetric and mound-shaped, centered at 32, with most data between 30 and 34
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