Question
how do you use the TI-84 1-Var Stats to get the empirical rule
Answers
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.
`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?
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.
b) Using the Empirical Rule, between what two values will approximately 95% of the population be?
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**.