Question
how do you use the TI-84 1-Var Stats to get the empirical rule
Answers
Answer
`x̄`=103
Ex = 21090
Ex^2 = 2340990
Sx = 6.03174603
`σ`x=6
n= 190
minX= 103
down arrow Qv1 = 108
(a) Using the Empirical Rule, approximately what percentage of the population values are between 97 and 109?
Ex = 21090
Ex^2 = 2340990
Sx = 6.03174603
`σ`x=6
n= 190
minX= 103
down arrow Qv1 = 108
(a) Using the Empirical Rule, approximately what percentage of the population values are between 97 and 109?
Answer
b) Using the Empirical Rule, between what two values will approximately 95% of the population be?
Answered by
GPT-4o mini
The TI-84 calculator provides a method for performing basic statistical analysis, including calculating key statistics from a data set. The 1-Var Stats function is useful for summarizing data, including mean, standard deviation, and more. The empirical rule, often referred to in the context of normally distributed data, states that:
- About 68% of the data falls within one standard deviation of the mean.
- About 95% of the data falls within two standard deviations of the mean.
- About 99.7% of the data falls within three standard deviations of the mean.
To use the TI-84 calculator to apply the empirical rule, follow these steps:
### 1. Enter Your Data
1. Turn on the calculator and press the **`STAT`** button.
2. Select **`1: Edit`** from the menu.
3. Enter your data values into **`L1`** (or any other list).
### 2. Calculate the Statistics
1. After entering your data, press the **`STAT`** button again.
2. Use the right arrow key to navigate to **`CALC`**.
3. Select **`1: 1-Var Stats`**.
4. If your data is in **`L1`**, you can just press **`ENTER`**; otherwise, specify the list (e.g., `L2`, etc.) by typing it name and press **`ENTER`**.
### 3. Review the Output
The calculator will display several statistics, including:
- **`x̄`** (mean)
- **`σ`** (standard deviation)
- **`n`** (number of data points)
### 4. Apply the Empirical Rule
- **Mean (\( \bar{x} \))**: This is the average of your data.
- **Standard Deviation (\( \sigma \))**: This tells you how spread out your data is.
Based on these two values, you can calculate the ranges for the empirical rule:
- **For 68% of the data**:
- Range: \( [\bar{x} - \sigma, \bar{x} + \sigma] \)
- **For 95% of the data**:
- Range: \( [\bar{x} - 2\sigma, \bar{x} + 2\sigma] \)
- **For 99.7% of the data**:
- Range: \( [\bar{x} - 3\sigma, \bar{x} + 3\sigma] \)
### 5. Interpret the Results
Once you've calculated the ranges, you can analyze how many of your data points fall within these ranges by simply counting them or using a histogram if you want a visual representation.
By following these steps, you can effectively use your TI-84 calculator to understand and apply the empirical rule to your data set.
- About 68% of the data falls within one standard deviation of the mean.
- About 95% of the data falls within two standard deviations of the mean.
- About 99.7% of the data falls within three standard deviations of the mean.
To use the TI-84 calculator to apply the empirical rule, follow these steps:
### 1. Enter Your Data
1. Turn on the calculator and press the **`STAT`** button.
2. Select **`1: Edit`** from the menu.
3. Enter your data values into **`L1`** (or any other list).
### 2. Calculate the Statistics
1. After entering your data, press the **`STAT`** button again.
2. Use the right arrow key to navigate to **`CALC`**.
3. Select **`1: 1-Var Stats`**.
4. If your data is in **`L1`**, you can just press **`ENTER`**; otherwise, specify the list (e.g., `L2`, etc.) by typing it name and press **`ENTER`**.
### 3. Review the Output
The calculator will display several statistics, including:
- **`x̄`** (mean)
- **`σ`** (standard deviation)
- **`n`** (number of data points)
### 4. Apply the Empirical Rule
- **Mean (\( \bar{x} \))**: This is the average of your data.
- **Standard Deviation (\( \sigma \))**: This tells you how spread out your data is.
Based on these two values, you can calculate the ranges for the empirical rule:
- **For 68% of the data**:
- Range: \( [\bar{x} - \sigma, \bar{x} + \sigma] \)
- **For 95% of the data**:
- Range: \( [\bar{x} - 2\sigma, \bar{x} + 2\sigma] \)
- **For 99.7% of the data**:
- Range: \( [\bar{x} - 3\sigma, \bar{x} + 3\sigma] \)
### 5. Interpret the Results
Once you've calculated the ranges, you can analyze how many of your data points fall within these ranges by simply counting them or using a histogram if you want a visual representation.
By following these steps, you can effectively use your TI-84 calculator to understand and apply the empirical rule to your data set.
Answered by
GPT-4o mini
To apply the Empirical Rule and find the percentage of the population values that are between 97 and 109, we first identify the mean (\( \bar{x} \)) and the standard deviation (\( \sigma \)) from the given data:
- Mean (\( \bar{x} \)) = 103
- Standard Deviation (\( \sigma \)) = 6
### Step 1: Identify the Range
Next, we calculate the corresponding values in relation to the mean and standard deviations:
- **Lower Bound**: \( \bar{x} - \sigma = 103 - 6 = 97 \)
- **Upper Bound**: \( \bar{x} + \sigma = 103 + 6 = 109 \)
### Step 2: Apply the Empirical Rule
The Empirical Rule states:
- Approximately **68%** of the data falls within one standard deviation of the mean.
- Therefore, the interval \( [97, 109] \) covers approximately **68%** of the population values.
### Conclusion
Thus, using the Empirical Rule, approximately **68%** of the population values are between 97 and 109.
- Mean (\( \bar{x} \)) = 103
- Standard Deviation (\( \sigma \)) = 6
### Step 1: Identify the Range
Next, we calculate the corresponding values in relation to the mean and standard deviations:
- **Lower Bound**: \( \bar{x} - \sigma = 103 - 6 = 97 \)
- **Upper Bound**: \( \bar{x} + \sigma = 103 + 6 = 109 \)
### Step 2: Apply the Empirical Rule
The Empirical Rule states:
- Approximately **68%** of the data falls within one standard deviation of the mean.
- Therefore, the interval \( [97, 109] \) covers approximately **68%** of the population values.
### Conclusion
Thus, using the Empirical Rule, approximately **68%** of the population values are between 97 and 109.
Answered by
GPT-4o mini
To find the range that contains approximately 95% of the population values using the Empirical Rule, we will extend our calculations to two standard deviations from the mean.
### Given:
- Mean (\( \bar{x} \)) = 103
- Standard Deviation (\( \sigma \)) = 6
### Step 1: Calculate Two Standard Deviations
1. **Lower Bound**:
\[
\bar{x} - 2\sigma = 103 - 2 \times 6 = 103 - 12 = 91
\]
2. **Upper Bound**:
\[
\bar{x} + 2\sigma = 103 + 2 \times 6 = 103 + 12 = 115
\]
### Conclusion
Using the Empirical Rule, approximately **95%** of the population values will be between **91 and 115**.
### Given:
- Mean (\( \bar{x} \)) = 103
- Standard Deviation (\( \sigma \)) = 6
### Step 1: Calculate Two Standard Deviations
1. **Lower Bound**:
\[
\bar{x} - 2\sigma = 103 - 2 \times 6 = 103 - 12 = 91
\]
2. **Upper Bound**:
\[
\bar{x} + 2\sigma = 103 + 2 \times 6 = 103 + 12 = 115
\]
### Conclusion
Using the Empirical Rule, approximately **95%** of the population values will be between **91 and 115**.
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