Population Parameters Common Core Algebra 2 Homework Answers

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How to Find Population Parameters Using Common Core Algebra 2

Population parameters are numerical values that describe the characteristics of a population, such as the mean, median, mode, standard deviation, and variance. In this article, we will show you how to use Common Core Algebra 2 concepts and skills to find population parameters from a given data set or a sample.

First, let's review some definitions:

A population is a set of all individuals or objects that share a common characteristic.

A sample is a subset of the population that is selected for observation or measurement.

A parameter is a numerical value that describes a population.

A statistic is a numerical value that describes a sample.

A sampling distribution is the distribution of all possible values of a statistic for a given sample size.

The central limit theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution with the same mean and standard deviation as the population.

To find population parameters, we can use two methods: direct calculation or inference from a sample.

Direct Calculation

If we have access to the entire population data, we can calculate the population parameters directly using formulas. For example, if we want to find the mean and standard deviation of the population, we can use these formulas:

The population mean is given by:

$\mu = \frac{\sum_{i=1}^N x_i}{N}$

The population standard deviation is given by:

$\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}$

where $x_i$ are the individual values in the population, $N$ is the population size, $\mu$ is the population mean, and $\sigma$ is the population standard deviation.

For example, suppose we have a population of 10 students and their test scores are: 80, 85, 90, 95, 100, 75, 70, 65, 60, 55. To find the population mean and standard deviation, we can use these formulas:

The population mean is:

$\mu = \frac{80 + 85 + 90 + 95 + 100 + 75 + 70 + 65 + 60 + 55}{10} = \frac{775}{10} = 77.5$

The population standard deviation is:

$\sigma = \sqrt{\frac{(80 - 77.5)^2 + (85 - 77.5)^2 + (90 - 77.5)^2 + (95 - 77.5)^2 + (100 - 77.5)^2 + (75 - 77.5)^2 + (70 - 77.5)^2 + (65 - 77.5)^2 + (60 - 77.5)^2 + (55 - 77.5)^2}{10}}$

$\sigma = \sqrt{\frac{156.25}{10}} = \sqrt{15.625} = 3.95$

Therefore, the population mean is $\mu = 77.5$ and the population standard deviation is $\sigma = 3.95$.

Inference from a Sample

If we do not have access to the entire population data, we can use a sample to estimate the population parameters using statistics and confidence intervals. For example, if we want to estimate the mean and standard deviation of the population, we can use these statistics:

The sample mean is given by:

$\overline{x} = \frac{\sum_{i=1}^n x_i}{n}$

The sample standard deviation is given by:

$s = \sqrt{\frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n-1}}$

where $x_i 061ffe29dd