Solved Statistics Worksheet #2: Normal Curves, Z-Scores, and ... - Free Printable

Here are the answers to your statistics worksheet.

Part 1: General Normal Distribution Questions

These questions ask about the properties of any standard normal curve (the bell shape).

1. Below the mean?
The mean is the exact center of the curve. Since the curve is perfectly symmetrical, exactly half of the data is on the left side.
Answer: 50%

2. Above the mean?
Just like the question above, the other half of the data is on the right side.
Answer: 50%

3. Within one standard deviation of the mean?
According to the Empirical Rule (68-95-99.7 rule), approximately 68% of all data falls within one step (standard deviation) of the center.
Answer: 68%

4. Within two standard deviations of the mean?
The Empirical Rule states that approximately 95% of data falls within two steps of the center.
Answer: 95%

5. Within three standard deviations of the mean?
The Empirical Rule states that approximately 99.7% of data falls within three steps of the center.
Answer: 99.7%

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Part 2: Biology Test Scores

Given Data:
* Mean ($\mu$) = 70
* Standard Deviation ($\sigma$) = 5

Labeling the Graph:
To fill in the blanks on the axis of the graph, we start at the mean (70) and add or subtract the standard deviation (5) for each mark.

* Center: 70
* Right +1: $70 + 5 = 75$
* Right +2: $75 + 5 = 80$
* Right +3: $80 + 5 = 85$
* Left -1: $70 - 5 = 65$
* Left -2: $65 - 5 = 60$
* Left -3: $60 - 5 = 55$

So the marks on the axis from left to right are: 55, 60, 65, 70, 75, 80, 85.

Calculating Percentages:
We use the Empirical Rule breakdown:
* From Mean to $\pm 1$ SD: 34% each side.
* From $\pm 1$ SD to $\pm 2$ SD: 13.5% each side.
* From $\pm 2$ SD to $\pm 3$ SD: 2.35% each side.
* Beyond $\pm 3$ SD: 0.15% each tail.

a) What percentage of scores are between scores 65 and 75?
* 65 is one standard deviation below the mean (-1).
* 75 is one standard deviation above the mean (+1).
* This covers the middle section.
* Calculation: $34\% + 34\% = 68\%$
* Answer: 68%

b) What percentage of scores are between scores 60 and 70?
* 60 is two standard deviations below the mean (-2).
* 70 is the mean.
* We need the area from -2 to 0. This is the sum of the slice from -2 to -1 (13.5%) and the slice from -1 to 0 (34%).
* Calculation: $13.5\% + 34\% = 47.5\%$
* Answer: 47.5%

c) What percentage of scores are between scores 60 and 85?
* 60 is two standard deviations below (-2).
* 85 is three standard deviations above (+3).
* We add up the slices:
* From -2 to -1: 13.5%
* From -1 to 0 (mean): 34%
* From 0 to +1: 34%
* From +1 to +2: 13.5%
* From +2 to +3: 2.35%
* Calculation: $13.5 + 34 + 34 + 13.5 + 2.35 = 97.35\%$
* Answer: 97.35%

d) What percentage of scores is less than a score of 55?
* 55 is three standard deviations below the mean (-3).
* "Less than" means the entire left tail.
* We know 99.7% is inside $\pm 3$ SDs. That leaves $100\% - 99.7\% = 0.3\%$ for both tails combined.
* One tail is half of that.
* Calculation: $0.3\% / 2 = 0.15\%$
* Answer: 0.15%

e) What percentage of scores is greater than a score of 80?
* 80 is two standard deviations above the mean (+2).
* "Greater than" means the right tail starting from +2.
* This includes the slice from +2 to +3 (2.35%) and the tiny tip beyond +3 (0.15%).
* Calculation: $2.35\% + 0.15\% = 2.5\%$
* Answer: 2.5%

f) Approximately how many biology students scored between 60 and 70?
* Total students = 2000.
* From question (b), we know the percentage between 60 and 70 is 47.5%.
* We need to find 47.5% of 2000.
* Calculation: $2000 \times 0.475 = 950$
* Answer: 950 students

g) Approximately how many biology students scored between 55 and 60?
* Total students = 2000.
* First, find the percentage. 55 is -3 SD and 60 is -2 SD.
* Looking at our breakdown, the slice between -3 and -2 is 2.35%.
* We need to find 2.35% of 2000.
* Calculation: $2000 \times 0.0235 = 47$
* Answer: 47 students

Final Answer:
Part 1:
1. 50%
2. 50%
3. 68%
4. 95%
5. 99.7%

Part 2:
Graph Labels (left to right): 55, 60, 65, 70, 75, 80, 85

a) 68%
b) 47.5%
c) 97.35%
d) 0.15%
e) 2.5%
f) 950 students
g) 47 students