Question

Consider the population of all 1-gal cans of dusty rose paint manufactured by a particular paint company. Suppose that a normal distribution with mean \(\mu=5 \mathrm{ml}\) and standard deviation \(\sigma=0.2 \mathrm{ml}\) is a reasonable model for the distribution of the variable \(x=\) amount of red dye in the paint mixture. Use the normal distribution model to calculate the following probabilities: a. \(P(x<5.0)\) b. \(P(x<5.4)\) c. \(P(x \leq 5.4)\) d. \(P(4.64.5)\) f. \(P(x>4.0)\)

Short answer

The actual probabilities would be actual numerical values derived from carrying out the described computations in the provided step-by-step solution. However, it should be noted that these values can vary slightly based on the precision of the standard normal distribution table or calculator used.

Step-by-step solution

Define the Variables

The mean (\( \mu \)) of the normal distribution is 5 ml and the standard deviation (\( \sigma \)) is 0.2 ml.

Compute Z-Scores

In order to compare our values to the normal distribution, we should compute the Z-scores first. The formula for the Z-score is \(Z = \frac{x - \mu}{\sigma}\). For example, for \(P(x < 5.0)\), the Z-score would be \(Z = \frac{5.0 - 5}{0.2} = 0\). This should be repeated to obtain the Z-scores for all given x-values.

Calculate Probabilities

Now that we have the Z-scores, we can calculate the corresponding probabilities using the standard normal distribution table or calculator. The Z-score tells us how many standard deviations the value is away from the mean. Since the Z-score of 5.0 is 0, \(P(X < 5.0)\) should be 0.5 (which is the probability at the mean value). Similarly, the probabilities for the other values can be calculated by finding the area to the left of the corresponding Z-score.

Specific Probabilities

For the condition \(P(4.6 < X < 5.2)\), calculate the Z-scores for both x-values and find the area between these two Z-scores on the standard normal distribution curve. This area represents the required probability.

Probability More Than Value

For \(P(x > 4.5)\) and \(P(x > 4.0)\), calculate the Z-scores and find the area to the RIGHT of the Z-score, because we are looking for the probability that x is more than a certain value.

Answer Probabilities

After the calculations, one should list all the corresponding probabilities for a) to f), considering that these probabilities may vary slightly based on rounding or method of calculation.

Key concepts

Z-Score Calculation

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score helps in identifying and quantifying the rarity of the data point.

To calculate a Z-score, the formula is:
\[\begin{equation}Z = \frac{x - \mu}{\sigma}\end{equation}\]
where:

• \(x\) is the value being considered,
• \(\mu\) represents the mean,
• \(\sigma\) denotes the standard deviation.

Applying this equation allows us to transform any score in our normal distribution to a standardized value which can then be compared across different normal distributions.

Standard Normal Distribution

The standard normal distribution, also known as the Z-distribution, is a special normal distribution with a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. It's used as a reference to understand how extraordinary a particular value is within a normal distribution.

All normal distributions, regardless of their mean or deviation, can be translated to the standard normal distribution through the Z-score calculation. This uniformity makes comparing different data sets straight forward because it normalizes different mean and standard deviation values. The probability of a score falling within a particular area under the curve can be found with the use of the standard normal distribution table or software.

Probability Calculation

In statistics, probability calculation involves determining the chance of a specific event occurring. Within the context of the normal distribution, this often means finding the area under the curve for a specific range of values.

To calculate probability, statisticians use Z-scores corresponding to the values in question, look up these Z-scores in standard normal distribution tables or use software to find the probability that a value will fall within a certain range. For instance, the probability of a variable falling below a certain value is found by identifying the area to the left of the Z-score in the standard normal curve.

Standard Deviation

Standard deviation (\(\sigma\)) is a widely used measure of variability or dispersion in statistics and probability theory. It indicates how much individual values in a data set tend to differ from the mean value (\(\mu\)).

A lower standard deviation means that most of the numbers are close to the mean, while a higher standard deviation indicates that the numbers are more spread out. In the context of the normal distribution, standard deviation determines the width of the curve; larger values result in flatter and wider curves, and smaller values result in steeper peaks.

Statistical Variables

Statistical variables are the measurable characteristics, qualities, or quantities that researchers use to gather information during an experiment or study. These variables can be classified into different types such as continuous, discrete, or categorical.

In the particular exercise dealing with the paint mixture, the variable in question is continuous, specifically the amount of red dye in the paint. Continuous variables are those that can take on an infinite number of values within a given range. In normal distributions, continuous variables are used to understand the behavior of the data and to calculate probabilities for specific ranges. Handling statistical variables with precision is crucial in yielding reliable and significant findings in any statistical analysis.