Population Parameters Common Core Algebra 2 Homework Answers
Julia Heaslet
Statistics has become the universal language of the sciences, and data analysis can lead to powerful results. As scientists, researchers, and managers working in the natural resources sector, we all rely on statistical analysis to help us answer the questions that arise in the populations we manage. For example:
These are typical questions that require statistical analysis for the answers. In order to answer these questions, a good random sample must be collected from the population of interests. We then use descriptive statistics to organize and summarize our sample data. The next step is inferential statistics, which allows us to use our sample statistics and extend the results to the population, while measuring the reliability of the result. But before we begin exploring different types of statistical methods, a brief review of descriptive statistics is needed.
Populations are characterized by descriptive measures called parameters. Inferences about parameters are based on sample statistics. For example, the population mean () is estimated by the sample mean (x̄). The population variance (σ2) is estimated by the sample variance (s2).
Variables are divided into two major groups: qualitative and quantitative. Qualitative variables have values that are attributes or categories. Mathematical operations cannot be applied to qualitative variables. Examples of qualitative variables are gender, race, and petal color. Quantitative variables have values that are typically numeric, such as measurements. Mathematical operations can be applied to these data. Examples of quantitative variables are age, height, and length.
Descriptive measures of populations are called parameters and are typically written using Greek letters. The population mean is μ (mu). The population variance is σ2 (sigma squared) and population standard deviation is σ (sigma).
Descriptive measures of samples are called statistics and are typically written using Roman letters. The sample mean is (x-bar). The sample variance is s2 and the sample standard deviation is s. Sample statistics are used to estimate unknown population parameters.
The population mean is represented by the Greek letter μ (mu). The sample mean is represented by x̄(x-bar). The sample mean is usually the best, unbiased estimate of the population mean. However, the mean is influenced by extreme values (outliers) and may not be the best measure of center with strongly skewed data. The following equations compute the population mean and sample mean.
The median of a variable is the middle value of the data set when the data are sorted in order from least to greatest. It splits the data into two equal halves with 50% of the data below the median and 50% above the median. The median is resistant to the influence of outliers, and may be a better measure of center with strongly skewed data.
The mode is the most frequently occurring value and is commonly used with qualitative data as the values are categorical. Categorical data cannot be added, subtracted, multiplied or divided, so the mean and median cannot be computed. The mode is less commonly used with quantitative data as a measure of center. Sometimes each value occurs only once and the mode will not be meaningful.
Understanding the relationship between the mean and median is important. It gives us insight into the distribution of the variable. For example, if the distribution is skewed right (positively skewed), the mean will increase to account for the few larger observations that pull the distribution to the right. The median will be less affected by these extreme large values, so in this situation, the mean will be larger than the median. In a symmetric distribution, the mean, median, and mode will all be similar in value. If the distribution is skewed left (negatively skewed), the mean will decrease to account for the few smaller observations that pull the distribution to the left. Again, the median will be less affected by these extreme small observations, and in this situation, the mean will be less than the median.
Measures of center look at the average or middle values of a data set. Measures of dispersion look at the spread or variation of the data. Variation refers to the amount that the values vary among themselves. Values in a data set that are relatively close to each other have lower measures of variation. Values that are spread farther apart have higher measures of variation.
Examine the two histograms below. Both groups have the same mean weight, but the values of Group A are more spread out compared to the values in Group B. Both groups have an average weight of 267 lb. but the weights of Group A are more variable.
The variance uses the difference between each value and its arithmetic mean. The differences are squared to deal with positive and negative differences. The sample variance (s2) is an unbiased estimator of the population variance (σ2), with n-1 degrees of freedom.
The standard deviation is the square root of the variance (both population and sample). While the sample variance is the positive, unbiased estimator for the population variance, the units for the variance are squared. The standard deviation is a common method for numerically describing the distribution of a variable. The population standard deviation is σ (sigma) and sample standard deviation is s.
We want to use this sample mean to estimate the true but unknown population mean. But our sample of 100 trees is just one of many possible samples (of the same size) that could have been randomly selected. Imagine if we take a series of different random samples from the same population and all the same size:
The sample mean (x̄) is a random variable with its own probability distribution called the sampling distribution of the sample mean. The distribution of the sample mean will have a mean equal to and a standard deviation equal to .
To compare standard deviations between different populations or samples is difficult because the standard deviation depends on units of measure. The coefficient of variation expresses the standard deviation as a percentage of the sample or population mean. It is a unitless measure.
Variability is described in many different ways. Standard deviation measures point to point variability within a sample, i.e., variation among individual sampling units. Coefficient of variation also measures point to point variability but on a relative basis (relative to the mean), and is not influenced by measurement units. Standard error measures the sample to sample variability, i.e. variation among repeated samples in the sampling process. Typically, we only have one sample and standard error allows us to quantify the uncertainty in our sampling process.
Data organization and summarization can be done graphically, as well as numerically. Tables and graphs allow for a quick overview of the information collected and support the presentation of the data used in the project. While there are a multitude of available graphics, this chapter will focus on a specific few commonly used tools.
Pie charts are a good visual tool allowing the reader to quickly see the relationship between categories. It is important to clearly label each category, and adding the frequency or relative frequency is often helpful. However, too many categories can be confusing. Be careful of putting too much information in a pie chart. The first pie chart gives a clear idea of the representation of fish types relative to the whole sample. The second pie chart is more difficult to interpret, with too many categories. It is important to select the best graphic when presenting the information to the reader.
Bar charts graphically describe the distribution of a qualitative variable (fish type) while histograms describe the distribution of a quantitative variable discrete or continuous variables (bear weight).
Boxplots use the 5-number summary (minimum and maximum values with the three quartiles) to illustrate the center, spread, and distribution of your data. When paired with histograms, they give an excellent description, both numerically and graphically, of the data.
With symmetric data, the distribution is bell-shaped and somewhat symmetric. In the boxplot, we see that Q1 and Q3 are approximately equidistant from the median, as are the minimum and maximum values. Also, both whiskers (lines extending from the boxes) are approximately equal in length.
Once we have organized and summarized your sample data, the next step is to identify the underlying distribution of our random variable. Computing probabilities for continuous random variables are complicated by the fact that there are an infinite number of possible values that our random variable can take on, so the probability of observing a particular value for a random variable is zero. Therefore, to find the probabilities associated with a continuous random variable, we use a probability density function (PDF).
Many continuous random variables have a bell-shaped or somewhat symmetric distribution. This is a normal distribution. In other words, the probability distribution of its relative frequency histogram follows a normal curve. The curve is bell-shaped, symmetric about the mean, and defined by and σ (the mean and standard deviation).
There are normal curves for every combination of and σ. The mean () shifts the curve to the left or right. The standard deviation (σ) alters the spread of the curve. The first pair of curves have different means but the same standard deviation. The second pair of curves share the same mean () but have different standard deviations. The pink curve has a smaller standard deviation. It is narrower and taller, and the probability is spread over a smaller range of values. The blue curve has a larger standard deviation. The curve is flatter and the tails are thicker. The probability is spread over a larger range of values.
There are millions of possible combinations of means and standard deviations for continuous random variables. Finding probabilities associated with these variables would require us to integrate the PDF over the range of values we are interested in. To avoid this, we can rely on the standard normal distribution. The standard normal distribution is a special normal distribution with a = 0 and σ = 1. We can use the Z-score to standardize any normal random variable, converting the x-values to Z-scores, thus allowing us to use probabilities from the standard normal table. So how do we find area under the curve associated with a Z-score?
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