Question
The following TI-84 Plus display presents some population parameters. Assume the distribution is bell-shaped.
x= 116
Ex = 14508
Ex^2= 1798992
Sx= 5.04310345
ox=5
n= 117
minX =118
down arrow Qv1 = 119
) Using the Empirical Rule, approximately what percentage of the population values are between 111 and 121?
x= 116
Ex = 14508
Ex^2= 1798992
Sx= 5.04310345
ox=5
n= 117
minX =118
down arrow Qv1 = 119
) Using the Empirical Rule, approximately what percentage of the population values are between 111 and 121?
Answers
Answer
Using the Empirical Rule, between what two values will approximately 95% of the population be?
Answered by
GPT-4o mini
The Empirical Rule, also known as the 68-95-99.7 rule, applies to bell-shaped (normal) distributions. It states that:
- Approximately 68% of the data fall within one standard deviation of the mean.
- Approximately 95% of the data fall within two standard deviations of the mean.
- Approximately 99.7% of the data fall within three standard deviations of the mean.
Given the information:
- Mean (\( \mu \)) \( x = 116 \)
- Standard deviation (\( \sigma \)) \( o_x = 5 \)
To find the percentage of population values between 111 and 121:
1. **Calculate the number of standard deviations from the mean**:
- The lower limit (111) is:
\[
z = \frac{111 - 116}{5} = \frac{-5}{5} = -1
\]
- The upper limit (121) is:
\[
z = \frac{121 - 116}{5} = \frac{5}{5} = 1
\]
2. **Apply the Empirical Rule**:
- According to the Empirical Rule, approximately 68% of the data falls within one standard deviation of the mean (between \( \mu - \sigma \) and \( \mu + \sigma \)).
Thus, the approximate percentage of the population values between 111 and 121 is **68%**.
- Approximately 68% of the data fall within one standard deviation of the mean.
- Approximately 95% of the data fall within two standard deviations of the mean.
- Approximately 99.7% of the data fall within three standard deviations of the mean.
Given the information:
- Mean (\( \mu \)) \( x = 116 \)
- Standard deviation (\( \sigma \)) \( o_x = 5 \)
To find the percentage of population values between 111 and 121:
1. **Calculate the number of standard deviations from the mean**:
- The lower limit (111) is:
\[
z = \frac{111 - 116}{5} = \frac{-5}{5} = -1
\]
- The upper limit (121) is:
\[
z = \frac{121 - 116}{5} = \frac{5}{5} = 1
\]
2. **Apply the Empirical Rule**:
- According to the Empirical Rule, approximately 68% of the data falls within one standard deviation of the mean (between \( \mu - \sigma \) and \( \mu + \sigma \)).
Thus, the approximate percentage of the population values between 111 and 121 is **68%**.
Answered by
GPT-4o mini
Using the Empirical Rule, we can determine the interval that contains approximately 95% of the population values in a bell-shaped distribution.
Given:
- Mean (\( \mu \)) = 116
- Standard deviation (\( \sigma \)) = 5
### To find the interval for approximately 95% of the population:
1. **Determine the range of values that are within two standard deviations from the mean**. According to the Empirical Rule, about 95% of the data falls within two standard deviations of the mean:
\[
\text{Lower limit} = \mu - 2\sigma = 116 - 2(5) = 116 - 10 = 106
\]
\[
\text{Upper limit} = \mu + 2\sigma = 116 + 2(5) = 116 + 10 = 126
\]
### Conclusion:
Approximately 95% of the population values will be between **106 and 126**.
Given:
- Mean (\( \mu \)) = 116
- Standard deviation (\( \sigma \)) = 5
### To find the interval for approximately 95% of the population:
1. **Determine the range of values that are within two standard deviations from the mean**. According to the Empirical Rule, about 95% of the data falls within two standard deviations of the mean:
\[
\text{Lower limit} = \mu - 2\sigma = 116 - 2(5) = 116 - 10 = 106
\]
\[
\text{Upper limit} = \mu + 2\sigma = 116 + 2(5) = 116 + 10 = 126
\]
### Conclusion:
Approximately 95% of the population values will be between **106 and 126**.
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