Normal Distributions-Empirical Rule. Interactive worksheet ... - Free Printable

Here are the solutions to the problems on the worksheet, using the Empirical Rule (also known as the 68-95-99.7 rule).

Problem 1

The Empirical Rule states:
* 68% of data falls within 1 standard deviation.
* 95% of data falls within 2 standard deviations.
* 99.7% of data falls within 3 standard deviations.

a) within 1 standard deviation: 68%
b) within 2 standard deviations: 95%
c) within 3 standard deviations: 99.7%

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Problem 2

Given: Mean ($\mu$) = 175, Standard Deviation ($\sigma$) = 37.

First, let's find the values for 1, 2, and 3 standard deviations from the mean:
* $1\sigma$: $175 - 37 = 138$ and $175 + 37 = 212$
* $2\sigma$: $175 - 74 = 101$ and $175 + 74 = 249$
* $3\sigma$: $175 - 111 = 64$ and $175 + 111 = 286$

Now we answer the questions based on these intervals:

a) Interval (138, 212): This is exactly $\pm 1$ standard deviation.
Answer: 68%

b) Interval (101, 249): This is exactly $\pm 2$ standard deviations.
Answer: 95%

c) Interval (64, 286): This is exactly $\pm 3$ standard deviations.
Answer: 99.7%

d) Outside interval (138, 212): Total is 100%. Inside is 68%.
$100\% - 68\% = 32\%$
Answer: 32%

e) Outside interval (101, 249): Total is 100%. Inside is 95%.
$100\% - 95\% = 5\%$
Answer: 5%

f) Outside interval (64, 286): Total is 100%. Inside is 99.7%.
$100\% - 99.7\% = 0.3\%$
Answer: 0.3%

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Problem 3

Given: Mean = 170 cm, Standard Deviation = 8 cm.

Let's map out the standard deviations:
* Mean: 170
* $1\sigma$: $170 - 8 = 162$ and $170 + 8 = 178$
* $2\sigma$: $170 - 16 = 154$ and $170 + 16 = 186$
* $3\sigma$: $170 - 24 = 146$ and $170 + 24 = 194$

a) Between 162 cm and 170 cm:
This is the area from $-1\sigma$ to the Mean. Since 68% is between $\pm 1\sigma$, half of that is on the left side.
$68\% / 2 = 34\%$
Answer: 34%

b) Between 170 cm and 186 cm:
This is the area from the Mean to $+2\sigma$. Since 95% is between $\pm 2\sigma$, half of that is on the right side.
$95\% / 2 = 47.5\%$
Answer: 47.5%

c) Between 178 cm and 186 cm:
This is the slice between $+1\sigma$ and $+2\sigma$.
Area from Mean to $+2\sigma$ is 47.5%. Area from Mean to $+1\sigma$ is 34%.
$47.5\% - 34\% = 13.5\%$
Answer: 13.5%

d) Less than 162 cm:
162 is $-1\sigma$. The area to the left of $-1\sigma$ includes the tail beyond 1 SD.
Area outside $\pm 1\sigma$ is $100\% - 68\% = 32\%$. Half of that is on the left.
$32\% / 2 = 16\%$
Answer: 16%

e) Less than 154 cm:
154 is $-2\sigma$. The area to the left of $-2\sigma$ is the lower tail.
Area outside $\pm 2\sigma$ is $100\% - 95\% = 5\%$. Half of that is on the left.
$5\% / 2 = 2.5\%$
Answer: 2.5%

f) Greater than 162 cm:
162 is $-1\sigma$. We want everything to the right of this point.
This includes the area from $-1\sigma$ to Mean (34%) plus the entire right half of the curve (50%).
$34\% + 50\% = 84\%$
Answer: 84%

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Problem 4

Given: Mean = 20,000 bottles, Standard Deviation = 2,000 bottles. Total days = 260.

Map the standard deviations:
* Mean: 20,000
* $1\sigma$: 18,000 and 22,000
* $2\sigma$: 16,000 and 24,000
* $3\sigma$: 14,000 and 26,000

a) Under 18,000 bottles:
18,000 is $-1\sigma$. We want the area to the left of $-1\sigma$.
Area outside $\pm 1\sigma$ is 32%. Half is on the left.
Percentage = 16%.
Days = $16\%$ of $260 = 0.16 \times 260 = 41.6$.
Rounding to nearest whole day: 42 days

b) Over 16,000 bottles:
16,000 is $-2\sigma$. We want the area to the right of $-2\sigma$.
This is the area from $-2\sigma$ to Mean (47.5%) plus the right half (50%).
Percentage = $47.5\% + 50\% = 97.5\%$.
Days = $97.5\%$ of $260 = 0.975 \times 260 = 253.5$.
Rounding to nearest whole day: 254 days

c) Between 18,000 and 24,000 bottles:
18,000 is $-1\sigma$. 24,000 is $+2\sigma$.
Area from $-1\sigma$ to Mean = 34%.
Area from Mean to $+2\sigma$ = 47.5%.
Total Percentage = $34\% + 47.5\% = 81.5\%$.
Days = $81.5\%$ of $260 = 0.815 \times 260 = 211.9$.
Rounding to nearest whole day: 212 days

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Problem 5

Given: Mean = 100, Standard Deviation = 15.

Map the standard deviations:
* Mean: 100
* $1\sigma$: 85 and 115
* $2\sigma$: 70 and 130
* $3\sigma$: 55 and 145

a) Approximately 68% of IQ scores lie between:
This corresponds to $\pm 1$ standard deviation.
$100 - 15 = 85$ and $100 + 15 = 115$.
Answer: 85 and 115

b) Approximately 95% of IQ scores lie between:
This corresponds to $\pm 2$ standard deviations.
$100 - 30 = 70$ and $100 + 30 = 130$.
Answer: 70 and 130

c) Approximately 99.7% of IQ scores lie between:
This corresponds to $\pm 3$ standard deviations.
$100 - 45 = 55$ and $100 + 45 = 145$.
Answer: 55 and 145

d) Percent less than 115:
115 is $+1\sigma$.
Area to the left of $+1\sigma$ = Area from Mean to $+1\sigma$ (34%) + Left Half (50%).
$34\% + 50\% = 84\%$.
Answer: 84%

e) Percent greater than 130:
130 is $+2\sigma$.
Area to the right of $+2\sigma$ is the upper tail.
Area outside $\pm 2\sigma$ is 5%. Half is on the right.
$5\% / 2 = 2.5\%$.
Answer: 2.5%

f) Percent less than 85:
85 is $-1\sigma$.
Area to the left of $-1\sigma$ is the lower tail.
Area outside $\pm 1\sigma$ is 32%. Half is on the left.
$32\% / 2 = 16\%$.
Answer: 16%

Final Answer:
1. a) 68% b) 95% c) 99.7%
2. a) 68% b) 95% c) 99.7% d) 32% e) 5% f) 0.3%
3. a) 34% b) 47.5% c) 13.5% d) 16% e) 2.5% f) 84%
4. a) 42 days b) 254 days c) 212 days
5. a) 85 and 115 b) 70 and 130 c) 55 and 145 d) 84% e) 2.5% f) 16%