Probabilities & Z-Scores w/ Graphing Calculator: Videos & Practice Problems

Using a TI-84 calculator simplifies finding probabilities from z-scores and vice versa. For probabilities less than a z-score, use the normalcdf function, setting bounds from negative infinity to the z-score. For probabilities between two z-scores, adjust the bounds accordingly. To find a z-score from a given probability, utilize the inverse norm function, entering the probability and selecting the appropriate tail. Understanding these functions enhances efficiency in statistical analysis, particularly in normal distributions.

Probability From Given Z-Scores - TI-84 (CE) Calculator

In statistical analysis, understanding how to find probabilities associated with z-scores is crucial, especially when dealing with normal distributions. A z-score represents the number of standard deviations a data point is from the mean. To find probabilities from z-scores, one can utilize a z-table or a graphing calculator, such as the TI-84, which simplifies the process significantly.

When using a calculator, the first step is to visualize the problem by sketching a graph. For instance, if you want to find the probability that a z-score is less than -0.81, you would draw a normal distribution curve, marking the z-score on the x-axis. The area to the left of this z-score represents the probability.

To calculate this using the TI-84, access the distribution menu by pressing 2nd followed by VARS. From there, select the normalcdf function for numerical output or shade normal for a graphical representation. For a probability that z is less than a specific value, set the lower bound to a very large negative number (e.g., -1E99) and the upper bound to the z-score in question, which in this case is -0.81. The mean (μ) is 0 and the standard deviation (σ) is 1 for a standard normal distribution, so these values remain unchanged. After entering these parameters, hitting the DRAW button will display the graph and the corresponding area, which represents the probability.

For example, the area calculated for z < -0.81 is approximately 0.209, matching the value obtained from the z-table.

In scenarios where you need to find the probability between two z-scores, such as between -1 and 1, the process is similar. Again, sketch the normal distribution curve, marking both z-scores. This time, when using the calculator, set the lower bound to -1 and the upper bound to 1. The calculator will then compute the area between these two z-scores, which represents the probability of a z-score falling within that range. For this example, the area calculated is approximately 0.6827, confirming the results from previous calculations.

Utilizing a graphing calculator not only streamlines the process of finding probabilities associated with z-scores but also reduces the likelihood of errors that can occur when using a z-table. This method is particularly beneficial for more complex problems involving multiple z-scores or when visual representation aids in understanding the distribution of data.

Sketch a graph to represent the probability, then use a calculator to find it.
$P\left(Z>1.14\right)$

• A. 0.1271
• B. 0.1271
• C. 0.8729
• D. 0.8729

Answer: (A)

Z-Scores From Given Probability - TI-84 (CE) Calculator

In statistics, understanding how to find a z-score from a given probability is essential, especially when working with the normal distribution. The z-score represents the number of standard deviations a data point is from the mean. When given a probability, you can use the inverse normal function on a TI-84 calculator to find the corresponding z-score.

To illustrate this process, consider a scenario where you have a probability that z is less than an unknown z-score, specifically \( P(Z < z) = 0.0853 \). This probability indicates a small area under the normal curve, suggesting that the z-score will be negative. To find the z-score, start by sketching a normal distribution graph, marking the area to the left of the unknown z-score. This visual representation helps in understanding the problem context.

Next, access the inverse normal function on your calculator by pressing the 2nd button followed by VARS, and select the invNorm option. Input the probability (0.0853), and for a standard normal distribution, set the mean (\( \mu \)) to 0 and the standard deviation (\( \sigma \)) to 1. Since the area is to the left, select the left tail option. After hitting Paste and then Enter, the calculator will output the z-score, which in this case is approximately -1.37. This aligns with the expectation of a negative value based on the graph.

In another example, suppose you need to find a z-score where the probability that z is greater than some unknown number is given as \( P(Z > z) = 0.3409 \). This indicates an area to the right of the z-score. Again, sketch the normal distribution, noting that this area is larger than the previous example, suggesting a positive z-score. Follow the same steps on the calculator: access the inverse normal function, input the probability (0.3409), and set \( \mu = 0 \) and \( \sigma = 1 \). However, since this probability represents the right tail, you must select the right tail option. The calculator will then provide a z-score of approximately 0.41, which is consistent with the expectation of a small positive number.

For users of the standard TI-84 (not the CE version), it’s important to note that the calculator defaults to assuming left tail probabilities. Therefore, when given a right tail probability, you must convert it by calculating \( 1 - P(Z > z) \) to find the corresponding left tail probability. In this case, you would compute \( 1 - 0.3409 = 0.6591 \) and input that into the calculator to find the z-score.

Understanding these steps and the functionality of the inverse normal function is crucial for accurately determining z-scores from probabilities, enhancing your statistical analysis skills.

Use a calculator to find the z–scores of the region shown in the standard normal distribution below.
A = 0.800

• A. $z=-0.26,z=0.26$
• B. $z=-0.25,z=0.25$
• C. $z=-0.805,z=0.805$
• D. $z=-1.28,z=1.28$

Answer: (D)

Here’s what students ask on this topic:

How do you find probabilities from z-scores using a TI-84 calculator?

To find probabilities from z-scores using a TI-84 calculator, you can use the normalcdf function. First, access the distribution menu by pressing the 'second' button followed by 'vars'. Select 'normalcdf' from the list. For a probability that z is less than a given value, set the lower bound to a very large negative number (approximating negative infinity) and the upper bound to the z-score. For example, if the z-score is -0.81, input -1E99 for the lower bound and -0.81 for the upper bound. Ensure that the mean (μ) is 0 and the standard deviation (σ) is 1 for a standard normal distribution. This function will calculate the area under the curve, representing the probability.

How can you find a z-score from a given probability using a TI-84 calculator?

To find a z-score from a given probability using a TI-84 calculator, use the inverse norm function. Access the distribution menu by pressing 'second' and 'vars', then select 'inverse norm'. Enter the probability value directly if you have a TI-84 Plus CE, and set the mean (μ) to 0 and standard deviation (σ) to 1. Choose the tail direction based on the problem: 'left' for probabilities less than a z-score, 'right' for probabilities greater than a z-score. If using a regular TI-84 Plus, convert right-tail probabilities to left-tail by subtracting from 1. The calculator will provide the corresponding z-score.

What is the difference between normalcdf and inverse norm functions on a TI-84 calculator?

The normalcdf function on a TI-84 calculator is used to find the probability associated with a range of z-scores, essentially calculating the area under the normal distribution curve. You input the lower and upper bounds of the z-scores to get the probability. In contrast, the inverse norm function is used to find the z-score corresponding to a given probability. You input the probability, and the calculator returns the z-score that marks the boundary of that probability area. Both functions are essential for statistical analysis involving normal distributions.

How do you set up window settings on a TI-84 calculator for graphing probabilities?

To set up window settings on a TI-84 calculator for graphing probabilities, first access the 'window' settings by pressing the 'window' button. Adjust the Xmin and Xmax to cover the range of z-scores you are interested in, typically from -3 to 3 for standard normal distributions. Set Ymin and Ymax to cover the probability range, usually from 0 to 1. Ensure the Xscl and Yscl are set to appropriate values for clear graph visibility. These settings help visualize the graph accurately, ensuring the probability area is correctly displayed.

How do you clear the graph on a TI-84 calculator after calculating probabilities?

To clear the graph on a TI-84 calculator after calculating probabilities, press the 'second' button followed by 'program' to access the 'draw' menu. Select 'ClrDraw' to clear the graph. This action removes the current graph, allowing you to start fresh for the next problem. Clearing the graph is essential to avoid confusion and ensure accurate representation of new calculations.