Standard Normal Distribution: Calculating Probabilities and Z-Scores Using the TI-84 Calculator

Standard Normal Distribution and the TI-84 Calculator

Probability from Given Z-Scores

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. Probabilities associated with z-scores can be found using graphing calculators such as the TI-84 Plus CE. This is useful for quickly determining the likelihood that a value falls within a certain range in a normal distribution.

• Z-score (z): The number of standard deviations a value is from the mean.

• Probability (P): The area under the standard normal curve corresponding to a range of z-scores.

Example Calculations:

• Finding P(z < -0.81): Use the calculator to find the probability to the left of z = -0.81. Result: 0.2090

• Finding P(-1.2 < z < 1): Use the calculator to find the probability between z = -1.2 and z = 1. Result: 0.8427

Calculator Steps (TI-84):

1. Press 2nd then VARS to access DISTR.

2. Select 2: normalcdf(.

3. Enter lower and upper bounds for z (e.g., normalcdf(-1.2, 1, 0, 1)).

4. Press ENTER to get the probability.

Formula:

• To find the probability between two z-scores: $P(a < z < b) = \int_{a}^{b} \frac{1}{\sqrt{2\pi}} e^{-z^2/2} dz$

Z-Scores from Given Probabilities

Sometimes, you are given a probability and need to find the corresponding z-score. The TI-84 calculator can be used for this purpose using the invNorm function.

• invNorm: Finds the z-score for a given cumulative probability (area to the left).

Example Calculations:

• Finding z for P(z < x) = 0.8931: Use invNorm(0.8931, 0, 1). Result: z = 1.23

• Finding z for P(z < x) = 0.3409: Use invNorm(0.3409, 0, 1). Result: z = -0.41

Calculator Steps (TI-84):

1. Press 2nd then VARS to access DISTR.

2. Select 3: invNorm(.

3. Enter the cumulative probability, mean (0), and standard deviation (1) (e.g., invNorm(0.8931, 0, 1)).

4. Press ENTER to get the z-score.

Formula:

• To find the z-score for a given cumulative probability $p$: $z = \text{invNorm}(p, 0, 1)$

Practice and Application

• Sketching the standard normal curve helps visualize the area (probability) being calculated.

• Use the calculator for efficiency and accuracy in finding probabilities and z-scores.

Table: TI-84 Calculator Functions for Standard Normal Distribution

Function	Purpose	Input	Output
normalcdf	Find probability between two z-scores	lower, upper, mean, std dev	Probability (area under curve)
invNorm	Find z-score for given cumulative probability	probability, mean, std dev	z-score

Example Application: If you want to know the probability that a value is less than z = 1.5, use normalcdf(-∞, 1.5, 0, 1) on the calculator. If you know the probability is 0.95 and want the corresponding z-score, use invNorm(0.95, 0, 1).