Problem 13 - Algebra 2 and Trigonometry

Question

In \(10-13,\) the mean and standard deviation of a normal distribution are given. Find each probability to the nearest hundredth. mean \(=8,\) standard deviation \(=1, P(7 \leq x)\)

Step-by-Step Solution

Answer

The probability is approximately 0.84.

Step 1 - Identify Parameters

The mean of the distribution is \( \mu = 8 \) and the standard deviation is \( \sigma = 1 \). We need to find the probability \( P(7 \leq x) \).

Step 2 - Convert to Standard Normal Variable

To find the probability using the standard normal distribution, we first convert \( x = 7 \) to a standard normal variable \( z \) using the formula: \[ z = \frac{x - \mu}{\sigma} = \frac{7 - 8}{1} = -1 \].

Step 3 - Look Up Z-Score Probability

Find the probability of \( z \geq -1 \) using the standard normal distribution table. A \( z \)-score of \(-1\) corresponds to a cumulative probability of approximately \(0.1587\). Since we want \( P(Z \geq -1) \), we calculate \( 1 - P(Z < -1) = 1 - 0.1587 = 0.8413 \).

Step 4 - Round to Nearest Hundredth

Round the probability to the nearest hundredth: \( 0.8413 \approx 0.84 \).

Key Concepts

Standard Deviation

The standard deviation is a crucial concept in statistics that measures the amount of variation or spread in a set of data values. It helps to determine how far individual data points deviate from the mean (average) of the data set. In simpler terms, if the data points are close to the mean, the standard deviation is low; if they are spread out, it is high.

For a normal distribution, most data points are expected to fall within one standard deviation of the mean. This is because the distribution is symmetric, and typical data sets follow a predictable pattern where data points cluster around the mean.

Calculating the standard deviation involves several steps:

• Find the mean of the data set.
• Subtract the mean from each data point to find the deviations.
• Square each deviation to eliminate negative values.
• Calculate the mean of these squared deviations.
• Take the square root of this average to obtain the standard deviation.

In our original exercise, a standard deviation of 1 shows us the data is not very spread from the mean, making predictions about values (like finding specific probabilities) more manageable.

Z-Score

The z-score is a standard way to measure how many standard deviations a data point is from the mean of the data set. It is particularly useful when comparing different data points within a distribution or between distributions.

In a normal distribution, the z-score can help determine the location of a specific value relative to the mean. For example, a z-score of 0 indicates the value is exactly at the mean, while a positive z-score indicates above the mean, and a negative one below the mean.

To calculate a z-score, follow this formula:\[ z = \frac{x - \mu}{\sigma} \]where:

• \( x \) is the data point,
• \( \mu \) is the mean of the distribution,
• \( \sigma \) is the standard deviation.

In the example provided, converting an x-value of 7 into the corresponding z-score (-1) helps us tap into the standard normal distribution's properties to find probabilities, which ultimately assists in making statistical inferences.

Probability Calculation

Probability calculations in the context of a normal distribution involve determining the likelihood of an event given the nature of the distribution.

This is often accomplished by translating the given problem into the language of the standard normal distribution through z-scores. Once you have a z-score, you can utilize standard normal distribution tables to obtain probability estimates.

Here's how you calculate the probability:

• Subtract the mean from the value of interest and divide by the standard deviation to get the z-score.
• Use a z-score table (or a statistical software package) to find the probability associated with this z-score.
• Adjust the probability as necessary (e.g., 1 minus the table value if you're interested in the probability of the z-score being greater rather than less).

In the example from the exercise, calculating \( P(7 \leq x) \) involved determining the z-score for 7, finding the corresponding probability from a z-table, and then doing a simple subtraction to obtain the probability of being greater than or equal to 7.

Cumulative Probability

Cumulative probability is the likelihood that a random variable is less than or equal to a particular value. Essentially, it adds up the probabilities of all outcomes up to and including the specified value.

In a normal distribution, cumulative probability is typically illustrated by the area under the curve to the left of a given z-score. This is practical because it allows one to quickly determine how much of the data set lies below a particular threshold.

To find cumulative probability:

• Convert the specified value to a z-score using the formula \( z = \frac{x - \mu}{\sigma} \).
• Use a standard normal distribution table to find the cumulative probability corresponding to that z-score.
• The table value gives you the probability up to that point; you can subtract this value from 1 to get the probability of being greater if needed.

In the exercise, after calculating a z-score of -1, we looked up the cumulative probability as 0.1587. Since we're interested in \( P(Z \geq -1) \), the final step involved subtracting 0.1587 from 1 to reflect the area to the right of this z-score.