Finding Probabilities, Z Values, and X Values with the Normal Distribution-Excel: Videos & Practice Problems

To find probabilities from z scores using Excel, the =NORM.S.DIST function is essential. This function applies to the standard normal distribution, which has a mean (μ) of 0 and a standard deviation (σ) of 1. When given a z score, this function calculates the cumulative probability, representing the probability that a value is less than the specified z score. The syntax is =NORM.S.DIST(z, TRUE), where z is the z score, and the argument TRUE specifies that the cumulative distribution function (CDF) is used, returning the left-tail probability. For example, to find the probability that a z score is less than -1.5, you would use =NORM.S.DIST(-1.5, TRUE), which yields approximately 0.07, indicating a 7% chance.

When dealing with a normal distribution that is not standard (i.e., where the mean and standard deviation differ from 0 and 1), Excel’s =NORM.DIST function is used. This function calculates the cumulative probability for any normal distribution with mean μ and standard deviation σ. The syntax is =NORM.DIST(x, μ, σ, TRUE), where x is the value of interest, μ is the population mean, σ is the population standard deviation, and TRUE again specifies the cumulative distribution. This function returns the probability that a random variable is less than or equal to x, which corresponds to the left-tail probability under the normal curve.

For example, if the baking time for cookies is normally distributed with a mean of 11 minutes and a standard deviation of 0.76 minutes, to find the probability that a batch takes 10 minutes or less, you would use =NORM.DIST(10, 11, 0.76, TRUE). This calculation results in approximately 0.09, meaning there is a 9% chance that a batch will bake in 10 minutes or less.

To find the probability of a value being greater than a certain number, such as the probability that baking time exceeds 12.5 minutes, the complement rule is applied. Since =NORM.DIST only calculates probabilities for values less than or equal to x, the probability of exceeding 12.5 minutes is found by subtracting the cumulative probability from 1. This is expressed as:

Using the example values, this becomes:

This indicates a 2% chance that the baking time exceeds 12.5 minutes.

Understanding how to use these Excel functions efficiently allows for quick and accurate calculation of probabilities related to both standard and non-standard normal distributions. The key is recognizing when to use =NORM.S.DIST for z scores and when to use =NORM.DIST for general normal distributions, as well as applying the complement rule to find probabilities for values greater than a given threshold.

The weights of granola bars produced by a snack company follow a normal distribution with a mean (μ) of 35 grams and a standard deviation (σ) of 2.7 grams. To find probabilities related to these weights, the NORM.DIST function is used, which calculates the cumulative distribution function (CDF) for a normal distribution with any mean and standard deviation, not just the standard normal distribution.

For example, to find the probability that a randomly selected granola bar weighs less than 33 grams, we calculate the left-tail probability using the formula:

\[P(X < 33) = \text{NORM.DIST}(33, 35, 2.7, \text{TRUE}) \approx 0.23\]

This means there is about a 23% chance that a granola bar weighs less than 33 grams.

To find the probability that a granola bar weighs more than 38 grams, we need the right-tail probability. Since NORM.DIST returns the left-tail probability, we subtract this value from 1:

\[P(X > 38) = 1 - P(X < 38) = 1 - \text{NORM.DIST}(38, 35, 2.7, \text{TRUE}) \approx 0.13\]

This indicates about a 13% chance of selecting a granola bar weighing more than 38 grams.

To find the probability that a granola bar weighs between 33 and 38 grams, we consider the total area under the normal curve (which equals 1) and subtract the probabilities of the two tails:

\[P(33 < X < 38) = 1 - P(X < 33) - P(X > 38) \approx 1 - 0.23 - 0.13 = 0.64\]

This shows there is approximately a 64% chance that a granola bar’s weight falls between 33 and 38 grams.

Understanding how to use the normal distribution and cumulative distribution function is essential for calculating probabilities involving normally distributed data with any mean and standard deviation. The key is recognizing when to use left-tail probabilities directly and when to subtract from one to find right-tail probabilities, as well as combining these to find probabilities within intervals.

Understanding how to convert probabilities into z scores and x values is essential when working with the normal distribution, especially using Excel functions. To find a z score from a given probability, the =NORM.S.INV function is used. This function requires a left tail probability as input, which represents the probability that a z score is less than a certain value. The output is the corresponding z score on the standard normal distribution, where the mean is zero and the standard deviation is one. If the problem provides a right tail probability or any other form, it must first be converted into a left tail probability before using =NORM.S.INV. For example, if the probability that a z score is below a certain value is 0.58, entering =NORM.S.INV(0.58) in Excel returns approximately 0.202, indicating that 58% of the distribution lies below this z score.

When dealing with normal distributions that are not standard—meaning the mean (μ) is not zero and the standard deviation (σ) is not one—the =NORM.INV function is appropriate. This function also takes a left tail probability as its first input but requires the mean and standard deviation of the distribution as the second and third inputs, respectively. This allows direct calculation of the x value corresponding to a given cumulative probability without manually converting to a z score first.

For instance, consider a scenario where song lengths are normally distributed with a mean of 3.56 minutes and a standard deviation of 0.32 minutes. To find the song length below which 33% of songs fall, you would use =NORM.INV(0.33, 3.56, 0.32), which yields approximately 3.419 minutes. This means there is a 33% chance a song is shorter than 3.419 minutes.

In cases where the probability given is a right tail probability, such as the chance a song is longer than a certain length, the complement rule is applied to convert it into a left tail probability. The complement rule states that the probability of an event is equal to one minus the probability of its complement. So, if there is a 0.2 probability that a song is longer than a certain length, the left tail probability is 1 − 0.2 = 0.8. Using this in Excel, the function would be =NORM.INV(1 - 0.2, 3.56, 0.32), resulting in approximately 3.829 minutes. This indicates a 20% chance that a song exceeds 3.829 minutes in length.

These Excel functions streamline the process of working with normal distributions by allowing direct conversion from probabilities to z scores or x values, provided the probabilities are expressed as left tail probabilities. Mastery of these functions enhances the ability to analyze data and interpret probabilities within the context of normal distributions efficiently.

The running time of a fitness trainer for a mile follows a normal distribution with a mean (μ) of 7.65 minutes and a standard deviation (σ) of 0.39 minutes. To analyze probabilities related to this distribution, it is essential to understand how to find specific time values corresponding to given probabilities using the inverse normal distribution function.

For example, if there is a 67% chance that a run is shorter than a certain time, this represents a left-tail probability of 0.67. To find the corresponding time value (x), the inverse normal function can be applied with inputs: the cumulative probability (0.67), the mean (7.65), and the standard deviation (0.39). This calculation yields \(x = 7.822\) minutes, meaning there is a 67% probability that the running time is less than 7.822 minutes.

Similarly, if there is a 45% chance that a run is longer than a certain time, this corresponds to a right-tail probability of 0.45. To use the inverse normal function, convert this to a left-tail probability by subtracting from 1, resulting in 0.55. Using the mean and standard deviation as before, the inverse normal function gives \(x = 7.699\) minutes. Thus, there is a 45% chance that the running time exceeds 7.699 minutes.

These calculations rely on the properties of the normal distribution and the use of the inverse cumulative distribution function, often denoted as \(x = \text{norm.inv}(p, \mu, \sigma)\), where \(p\) is the cumulative probability, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. This approach allows for determining specific values of running time associated with given probabilities, which is crucial for interpreting and predicting outcomes in normally distributed data.