Question

A normal random variable \(x\) has mean \(\mu=1.20\) and standard deviation \(\sigma=.15 .\) Find the probabilities of these \(x\) -values: a. \(1.001.38\) c. \(1.35

Short answer

a) P(1.00 < x < 1.10) ≈ 0.2088 b) P(x > 1.38) ≈ 0.1151 c) P(1.35 < x < 1.50) ≈ 0.1359

Step-by-step solution

Convert \(x\)-values to \(Z\)-scores

First, let's convert the given \(x\)-values to their corresponding \(Z\)-scores using the formula: \(Z = \frac{x - \mu}{\sigma}\) For each range, we will find the \(Z\)-scores for both the lower and upper limits. a. \(1.00 < x < 1.10\): \(Z1 = \frac{1.00 - 1.20}{0.15} = -1.33\) \(Z2 = \frac{1.10 - 1.20}{0.15} = -0.67\) b. \(x > 1.38\): \(Z = \frac{1.38 - 1.20}{0.15} = 1.20\) c. \(1.35 < x < 1.50\): \(Z1 = \frac{1.35 - 1.20}{0.15} = 1.00\) \(Z2 = \frac{1.50 - 1.20}{0.15} = 2.00\)

Find probabilities for each range

Now, we will look up the probabilities in the standard normal distribution table or use a calculator. a. \(1.00 < x < 1.10\): The probability is the area between \(Z1\) and \(Z2\). In this case, \(P(-1.33 < Z < -0.67)\). b. \(x > 1.38\): The probability is the area to the right of the \(Z\)-score. In this case, \(P(Z > 1.20)\). c. \(1.35 < x < 1.50\): The probability is the area between \(Z1\) and \(Z2\). In this case, \(P(1.00 < Z < 2.00)\).

Calculate the probabilities

Use the standard normal distribution table or calculator to find the probabilities: a. \(P(-1.33 < Z < -0.67) \approx 0.2088\) b. \(P(Z > 1.20) \approx 0.1151\) c. \(P(1.00 < Z < 2.00) \approx 0.1359\) The probabilities for each range of \(x\) are: a. \(P(1.00 < x < 1.10) \approx 0.2088\) b. \(P(x > 1.38) \approx 0.1151\) c. \(P(1.35 < x < 1.50) \approx 0.1359\)

Key concepts

Mean

The mean is a crucial concept in statistics, especially when discussing the normal distribution. It represents the average of a set of numbers. In a normal distribution, the mean is the center point of the data. Here, the mean (\(\mu\)) is given as 1.20.

This value tells us where the middle of our data is located on the number line. In practical terms, this is the point around which the data is symmetrically distributed. Understanding the mean helps us gauge the overall level of a dataset.

• For example, if many data points are close to the mean, it suggests a concentration of values near this central point.
• If the mean changes, it implies a shift in the distribution of data points.

This concept is foundational, aiding in our understanding of how data points relate to each other.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a dataset are around the mean. In essence, it quantifies the variation within the data. A smaller standard deviation implies that the data points are close to the mean, whereas a larger standard deviation indicates more spread.

Given the standard deviation (\(\sigma\)) as 0.15 in our problem, it tells us how far the data points tend to be from the mean on average. Mathematically, the standard deviation is the square root of the variance, providing a sense of dispersion.

• If standard deviation is low, it means most numbers are close to the average.
• High standard deviation suggests the numbers are more spread out.

In our context, understanding the standard deviation helps in predicting the data's behavior and estimating probabilities when using the normal distribution table.

Z-scores

Z-scores are a way of standardizing scores on the same scale by converting an individual data point's deviation from the mean into standard deviations. The formula for calculating a Z-score is \(Z = \frac{x - \mu}{\sigma}\). This allows us to determine how far a specific score is from the mean, in units of standard deviations.

Z-scores are critical because they enable comparison between different sets of data by standardizing them.

• A positive Z-score indicates the data point is above the mean.
• A negative Z-score indicates the data point is below the mean.

In our example, we calculated multiple Z-scores for different ranges of x-values. These Z-scores were then used to find probabilities using the standard normal distribution, which is crucial for making statistical inferences.

Probability

Probability measures the likelihood of a specific event occurring. In the context of a normal distribution, it refers to finding the area under the curve, which corresponds to the likelihood of a range of values.

Using Z-scores, we translated x-values into standard terms, enabling us to utilize the standard normal distribution table to calculate these probabilities.

• The area between two Z-scores gives the probability of a value falling between those points.
• The area to the right or left of a Z-score can determine the probability of a value being greater or lesser than a certain point.

In our exercise, probabilities such as \(P(-1.33 < Z < -0.67)\) provide the chances of x-values being in a specific range. Understanding these probabilities helps in making predictions and informed decisions based on data.