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how do you use the TI-84 1-Var Stats to get the empirical rule

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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.

`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?

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.

b) Using the Empirical Rule, between what two values will approximately 95% of the population be?

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.