Problem 19 - Basic Statistics for Business and Economics

Question

A cola-dispensing machine is set to dispense on average 7.00 ounces of cola per cup. The standard deviation is 0.10 ounces. The distribution amounts dispensed follows a normal distribution. a. What is the probability that the machine will dispense between 7.10 and 7.25 ounces of cola? b. What is the probability that the machine will dispense 7.25 ounces of cola or more? c. What is the probability that the machine will dispense between 6.80 and 7.25 ounces of cola?

Step-by-Step Solution

Answer

a) 0.1525 b) 0.0062 c) 0.9710

Step 1: Understand the Problem

We are given that the cola dispensing machine dispenses amounts following a normal distribution with a mean \( \mu = 7.00 \) ounces and a standard deviation \( \sigma = 0.10 \) ounces. We need to find the probabilities for different specified ranges of dispensed amounts.

Step 2: Convert to Z-scores

Use the formula for the Z-score: \( Z = \frac{X - \mu}{\sigma} \). Convert the specified dispensing amounts into Z-scores.- For 7.10 ounces: \( Z = \frac{7.10 - 7.00}{0.10} = 1 \).- For 7.25 ounces: \( Z = \frac{7.25 - 7.00}{0.10} = 2.5 \).- For 6.80 ounces: \( Z = \frac{6.80 - 7.00}{0.10} = -2 \).

Step 3: Find Z-score Probabilities from Normal Distribution Table

Use the standard normal distribution table to find the probabilities for the Z-scores. - Probability that Z < 1: Approximate cumulative probability is 0.8413. - Probability that Z < 2.5: Approximate cumulative probability is 0.9938. - Probability that Z < -2: Approximate cumulative probability is 0.0228.

Step 4: Calculate Probability for Part (a)

To find the probability that the machine dispenses between 7.10 and 7.25 ounces:\( P(1 < Z < 2.5) = P(Z < 2.5) - P(Z < 1) = 0.9938 - 0.8413 = 0.1525 \).

Step 5: Calculate Probability for Part (b)

To find the probability that the machine dispenses 7.25 ounces or more:\( P(Z > 2.5) = 1 - P(Z < 2.5) = 1 - 0.9938 = 0.0062 \).

Step 6: Calculate Probability for Part (c)

To find the probability that the machine dispenses between 6.80 and 7.25 ounces:\( P(-2 < Z < 2.5) = P(Z < 2.5) - P(Z < -2) = 0.9938 - 0.0228 = 0.971 \).

Key Concepts

Z-score calculation

To understand how a cola-dispensing machine's output can be analyzed, we first need to delve into the concept of a Z-score. A Z-score, in statistics, is a measure that describes a value's position relative to the mean of a group of values. It is expressed in terms of standard deviations from the mean.Here's how you calculate a Z-score:- Use the formula: \[ Z = \frac{X - \mu}{\sigma} \] Where: - \(X\) is the value you are reporting - \(\mu\) is the mean of the distribution - \(\sigma\) is the standard deviationFor the cola machine, different Z-scores help us find the probabilities related to dispensing amounts. For example, to find the Z-score for the value 7.10 ounces:- Calculate: \[ Z = \frac{7.10 - 7.00}{0.10} = 1 \]This conversion allows us to look up probabilities in the standard normal distribution table, which tells us what proportion of data falls below (or above) this Z-score. Remember, Z-score conversion makes it easier to calculate probabilities for values in any normally distributed dataset.

Probability in statistics

Probability is the language of uncertainty in statistics, helping us quantify the likelihood of events. When dealing with a normal distribution, such as the amount of cola dispensed by a machine, probabilities give insights about the chance of certain outcomes occurring.To tackle probability calculations:- We use the Z-scores derived from our data points.- Look up these Z-scores in the standard normal distribution table.- The table provides cumulative probabilities, which show the chance that a value is less than or equal to a given Z-score.For example:- To find the probability that the machine dispenses between 7.10 and 7.25 ounces, calculate the difference between the cumulative probabilities at Z = 1 and Z = 2.5: \[ P(1 < Z < 2.5) = P(Z < 2.5) - P(Z < 1) = 0.9938 - 0.8413 = 0.1525 \]The result, 0.1525, indicates a 15.25% probability.Probability helps make informed decisions about where our data points are likely to be, based on historical data.

Standard deviation

Standard deviation is a crucial concept in statistics, representing the amount of variation or dispersion in a set of values. In simpler terms, it tells us how much the values in a dataset typically differ from the mean. Why is standard deviation important? - It gives a common measure of consistency in a dataset. - For the cola machine setup, the standard deviation of 0.10 ounces indicates the variability around the mean amount of 7.00 ounces. - A smaller standard deviation indicates the data points are close to the mean, while a larger one suggests more spread out data points. Calculating Z-scores uses standard deviation, as it scales the difference between a value and the mean into a standardized unit. This scaling is vital to apply the concept of a normal distribution uniformly, regardless of the original scale of data units.