Data Analysis and Probability - Part 2: Permutations, Binomial Distribution, Normal Distribution, and Hypothesis Testing

Algebra 2 by Holt McDougal
Algebra 2 by Holt McDougal

I already wrote Data Analysis and Probability - Part 1, which referenced Algebra 1 by Holt McDougal. This time, the reference/inspiration/influence comes from Algebra 2 by Holt McDougal: Chapter 7, Probability, and Chapter 8, Data Analysis and Statistics. I summarized them and gave the best examples to make them easy to understand. But, in this part, I don't cover what I already covered in Part 1 (to avoid repetition). Of course, it is better if you read the sources, Algebra 1 & 2 by Holt McDougal; they are great books.

Probability

In Algebra 2, probability builds on earlier concepts by introducing permutations, combinations, two-way tables, and compound events. These tools help us count outcomes efficiently and calculate probabilities in more complex situations. Understanding these foundational ideas prepares us to model real-world uncertainty using structured probability methods.

Permutations and Combinations

In Algebra 2, counting methods become very important when solving probability problems. Two major counting techniques are permutations and combinations. Both are used to count the number of possible outcomes, but the key difference is whether order matters.

Permutations

A permutation is an arrangement of objects in which order matters. If changing the order creates a different outcome, then you are dealing with a permutation.

The number of permutations of n objects taken r at a time is given by:

\[{}_n \mathrm{P}_r = \frac{n!}{(n-r)!}\]

The symbol n! (n factorial) means:

\[n! = n \cdot (n-1) \cdot (n-2) \cdots 3 \cdot 2 \cdot 1\]

Example:

How many different ways can 3 students be chosen from 5 students to receive first, second, and third place awards?

Because first, second, and third place are different positions, order matters. So we use permutations.

\[{}_5 \mathrm{P}_3 = \frac{5!}{(5-3)!}\]

\[= \frac{5!}{2!}\]

\[= \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{2 \cdot 1}\]

\[= 5 \cdot 4 \cdot 3 = 60\]

There are 60 different possible arrangements.

Combinations

A combination is a selection of objects in which order does not matter. If changing the order does not create a new outcome, then you are dealing with a combination.

The number of combinations of n objects taken r at a time is given by:

\[{}_n \mathrm{C}_r = \frac{n!}{r!(n-r)!}\]

Example:

How many ways can 3 students be chosen from 5 students to form a committee?

Because the order of the students does not matter, we use combinations.

\[{}_5 \mathrm{C}_3 = \frac{5!}{3!2!}\]

\[= \frac{5 \cdot 4 \cdot 3!}{3!2 \cdot 1}\]

\[= \frac{5 \cdot 4}{2} = 10\]

There are 10 possible committees.

The main difference between permutations and combinations is simple:

Type Order Matters? Formula
Permutation Yes \(\Large \frac{n!}{(n-r)!}\)
Combination No \(\Large \frac{n!}{r!(n-r)!}\)

Two-Way Tables

A two-way table displays data that can be classified according to two different categories. Two-way tables help organize data and make it easier to calculate probabilities involving two events.

Here is an example.

A survey of 100 students asked whether they prefer Math or Science and whether they are in Grade 10 or Grade 11.

Math Science Total
Grade 10 28 22 50
Grade 11 32 18 50
Total 60 40 100

From the table, we can calculate probabilities.

Example 1: What is the probability that a randomly selected student prefers Math?

\[P(\text{Math}) = \frac{60}{100} = 0.6\]

Example 2: What is the probability that a student is in Grade 10 and prefers Science?

\[P(\text{Grade 10 and Science}) = \frac{22}{100} = 0.22\]

Example 3: What is the probability that a student prefers Science given that the student is in Grade 11?

\[P(\text{Science | Grade 11}) = \frac{18}{50} = 0.36\]

Notice that conditional probability restricts the sample space to a specific row or column.

Compound Events

A compound event consists of two or more simple events. Compound events often use the words and or or.

In probability notation:

Probability of A and B

If events A and B are independent, then:

\[P(A \cap B) = P(A) \cdot P(B)\]

Example:

What is the probability of rolling a 4 on a number cube and flipping heads on a coin?

\[P(4) = \frac{1}{6}\]

\[P(H) = \frac{1}{2}\]

\[P(4 \cap H) = \frac{1}{6} \cdot \frac{1}{2}\]

\[= \frac{1}{12}\]

The probability of rolling a 4 and flipping heads is \(\frac{1}{12}\).

Probability of A or B (Mutually Exclusive Events)

If events A and B are mutually exclusive (they cannot occur at the same time), then:

\[P(A \cup B) = P(A) + P(B)\]

Example:

What is the probability of rolling a 2 or a 5 on a number cube?

\[P(2) = \frac{1}{6}\]

\[P(5) = \frac{1}{6}\]

\[P(2 \cup 5) = \frac{1}{6} + \frac{1}{6}\]

\[= \frac{2}{6} = \frac{1}{3}\]

The probability of rolling a 2 or a 5 is \(\frac{1}{3}\).

General Addition Rule

If events A and B are not mutually exclusive, then:

\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

This formula subtracts the overlap so the shared outcomes are not counted twice.

Example:

In a class of 30 students, 18 students play soccer, 12 students play basketball, and 5 students play both sports. What is the probability that a randomly selected student plays soccer or basketball?

\[P(\text{Soccer}) = \frac{18}{30}\]

\[P(\text{Basketball}) = \frac{12}{30}\]

\[P(\text{Soccer} \cap \text{Basketball}) = \frac{5}{30}\]

Use the General Addition Rule.

\[P(\text{Soccer} \cup \text{Basketball}) = \frac{18}{30} + \frac{12}{30} - \frac{5}{30}\]

\[= \frac{25}{30}\]

\[= \frac{5}{6}\]

The probability that a student plays soccer or basketball is \(\frac{5}{6}\).

Data Analysis and Statistics

Data analysis and statistics in Algebra 2 extend probability into modeling and inference. In this section, we study expected value, variance, standard deviation, sampling methods, hypothesis testing, binomial distributions, and normal distributions. These concepts allow us to analyze patterns in data, model random processes, and determine whether results are likely due to chance or represent meaningful trends in a population.

Measures of Central Tendency and Variation

In Part 1, we discussed mean, median, mode, range, quartiles, box-and-whisker plots, and outliers. In Algebra 2, we go deeper by studying expected value, variance, and standard deviation. These measures help describe how data are distributed and how far values spread from the mean.

Expected Value

The expected value of a random variable is the weighted average of all possible values. It represents the long-term average outcome of a probability experiment.

The formula for expected value is:

\[E(X) = \sum x \cdot P(x)\]

This means multiply each outcome by its probability, then add the results.

Example:

A game has the following outcomes:

Outcome (x) Probability P(x)
$10 0.2
$5 0.3
$0 0.5

Find the expected value.

\[E(X) = 10(0.2) + 5(0.3) + 0(0.5)\]

\[= 2 + 1.5 + 0\]

\[= 3.5\]

The expected value is $3.50.

Variance and Standard Deviation (Four-Step Process)

The variance and standard deviation measure how spread out data values are from the mean. Algebra 2 presents a clear four-step process to find them.

Step 1: Find the mean of the data.

\[\bar{x} = \frac{\text{sum of data values}}{\text{number of data values}}\]

Step 2: Find the difference between the mean and each data value, then square each difference.

Step 3: Find the variance by adding the squared differences and dividing by the number of data values.

\[\sigma^2 = \frac{\sum (x - \bar{x})^2}{n}\]

Step 4: Find the standard deviation by taking the square root of the variance.

\[\sigma = \sqrt{\sigma^2}\]

Example:

Find the variance and standard deviation of the data set: 4, 6, 8, 10.

Step 1: Find the mean.

\[\bar{x} = \frac{4 + 6 + 8 + 10}{4}\]

\[= \frac{28}{4}\]

\[= 7\]

Step 2: Find each difference from the mean and square it.

x x - 7 (x - 7)2
4 -3 9
6 -1 1
8 1 1
10 3 9

Step 3: Find the variance.

\[\sigma^2 = \frac{9 + 1 + 1 + 9}{4}\]

\[= \frac{20}{4}\]

\[= 5\]

The variance is 5.

Step 4: Find the standard deviation.

\[\sigma = \sqrt{5}\]

\[\approx 2.24\]

The standard deviation is approximately 2.24.

Data Gathering

To analyze data correctly, you must first understand how the data are collected. In Algebra 2, data gathering begins with identifying the population, sample, and whether the sample is unbiased.

Population, Census, and Sample

A population is the entire group of individuals or objects that you want information about.

A census is a survey that collects data from every member of the population.

A sample is a subset of the population.

A random sample is a sample in which every member of the population has an equal chance of being selected.

Example 1:

A school principal wants to know the average number of hours students spend on homework each week. She surveys 50 students chosen at random from the entire school.

Biased Samples

A sample is biased if some members of the population are more likely to be chosen than others.

Two common types of biased samples are:

Example 2:

A survey about school lunch preferences is conducted by asking students in the soccer team during practice.

This sample is biased because students who do not play soccer are underrepresented, and soccer players are overrepresented.

Statistic and Parameter

A parameter is a numerical measure that describes a population.

A statistic is a numerical measure that describes a sample.

Example 3: Analyzing a Survey

A random sample of 100 students from a school of 1,000 students shows that 60% prefer online homework.

Making Predictions

Statistics are often used to estimate parameters.

Example 4:

Using the previous survey result, predict how many students in the entire school prefer online homework.

\[0.60 \times 1000 = 600\]

Based on the sample statistic, we predict that approximately 600 students in the population prefer online homework.

This prediction is reasonable because the sample was random.

Surveys, Experiments, and Observational Studies

Data Collection Methods

There are two important data collection methods in Algebra 2: experiments and observational studies.

An experiment imposes a treatment on individuals in order to measure their responses.

An observational study observes individuals and measures variables without controlling or changing anything.

Example:

A researcher studies whether drinking an energy drink improves math test scores.

Types of Experiments

There are two main types of experiments discussed in Algebra 2.

Controlled Experiment

A controlled experiment compares a treatment group with a control group. The control group does not receive the treatment.

Example:

A scientist tests whether a new fertilizer helps plants grow taller.

The scientist compares the heights of the two groups.

Randomized Comparative Experiment

A randomized comparative experiment randomly assigns individuals to different treatment groups.

Random assignment helps reduce bias.

Example:

A medical study compares two headache medicines. Patients are randomly assigned to receive either Medicine A or Medicine B. The effectiveness of the medicines is then compared.

Because patients were randomly assigned, this is a randomized comparative experiment.

Designing an Experiment or Observational Study

Example:

A school wants to determine whether listening to music while studying improves test scores.

This research question is best addressed with an experiment, because we want to determine cause and effect.

How to set up the experiment:

If the school only observed students who already listen to music while studying, it would be an observational study instead.

Evaluating Data Collection Methods

When evaluating a data collection method, consider the following questions:

Example:

A study claims that students who sleep more score higher on tests. The researcher surveys honors students and records how many hours they sleep.

This study may be flawed because:

Therefore, conclusions from the study should be interpreted carefully.

Significance of Experimental Results

When conducting an experiment, it is important to determine whether the results are meaningful or simply due to chance. This process is called hypothesis testing.

Hypothesis Testing

A hypothesis is a claim about a population parameter.

The null hypothesis, written as \(H_0\), is the statement that there is no difference, no effect, or no change.

The null hypothesis assumes that any observed difference is due to random chance.

Researchers test whether there is enough evidence to reject the null hypothesis.

Example 1: Analyzing a Controlled Experiment

A company claims that a new energy drink improves reaction time. To test this claim, 40 participants are randomly divided into two groups:

The average reaction time of the treatment group is faster than the control group.

Step 1: State the null hypothesis.

\[H_0: \text{The energy drink has no effect on reaction time.}\]

Step 2: Analyze the results.

If the difference in reaction times is small, it may be due to chance.

If the difference is large and unlikely to occur by chance, the null hypothesis may be rejected.

Rejecting the null hypothesis suggests that the energy drink may improve reaction time.

Example 2: Using a z-Test

A z-test is used to determine whether a sample mean differs significantly from a known population mean.

The test statistic is:

\[\Large z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}\]

Where:

Example:

A cereal company claims that the average weight of its cereal boxes is 16 ounces. A random sample of 25 boxes has a mean weight of 15.5 ounces. The population standard deviation is known to be 1 ounce. Test the claim.

Step 1: State the null hypothesis.

\[H_0: \mu = 16\]

Step 2: Calculate the z-score.

\[z = \frac{15.5 - 16}{\frac{1}{\sqrt{25}}}\]

\[= \frac{-0.5}{\frac{1}{5}}\]

\[= \frac{-0.5}{0.2}\]

\[= -2.5\]

Step 3: Interpret the result.

A z-score of -2.5 means the sample mean is 2.5 standard errors below the population mean.

If the probability of obtaining a z-score this extreme is very small (typically less than 0.05), we reject the null hypothesis.

This suggests that the true mean weight may be different from 16 ounces.

Sampling Distributions

When studying a population, researchers often collect data from a sample. The way a sample is selected affects how reliable the conclusions are.

Types of Samples

A simple random sample is a sample in which every member of the population has an equal chance of being selected.

A systematic sample selects members at regular intervals from an ordered list.

A stratified sample divides the population into similar groups (strata) and randomly selects members from each group.

A cluster sample divides the population into groups (clusters) and randomly selects entire groups.

A convenience sample consists of individuals who are easily accessible.

A self-selected sample consists of individuals who volunteer to participate.

Simple random, systematic, stratified, and cluster samples are called probability samples because each member of the population has a known chance of being selected.

Example 1: Classifying a Sample

A school has 800 students. A researcher assigns each student a number and uses a random number generator to select 50 students.

This is a simple random sample because every student has an equal chance of being chosen.

This is also a probability sample.

If instead the researcher selected every 16th student from an alphabetical list, the sample would be systematic.

If the researcher divided students by grade level and randomly selected 10 students from each grade, the sample would be stratified.

If the researcher randomly selected two entire classrooms and surveyed all students in those rooms, the sample would be cluster.

Example 2: Evaluating Sampling Methods

A news station surveys viewers by asking people to respond to an online poll.

This is a self-selected sample, which may be biased because only people with strong opinions are likely to respond.

Suppose the survey reports that 52% of 1,000 randomly selected voters support a candidate, with a margin of error of 3%.

The margin of error measures how much the sample statistic is expected to differ from the population parameter.

The interval estimate is:

\[52\% \pm 3\%\]

\[= (49\%, 55\%)\]

This means the true population percentage is likely between 49% and 55%.

Example 3: Interpreting a Margin of Error

A survey states that 40% of students prefer online learning, with a margin of error of 4%.

This means the actual population percentage is likely between:

\[40\% - 4\% = 36\%\]

\[40\% + 4\% = 44\%\]

We can say with the stated level of confidence that between 36% and 44% of all students prefer online learning.

A smaller margin of error indicates a more precise estimate.

Binomial Distributions

A binomial distribution models the probability of getting a certain number of successes in a fixed number of independent trials. Each trial has only two possible outcomes: success or failure.

Binomial expansions and binomial probabilities are closely related. Pascal’s Triangle helps generate binomial coefficients used in expansions.

Pascal’s Triangle and Binomial Coefficients

Row Pascal's Triangle Combinations Binomial Expansion
0 1 \({}_0 \mathrm{C}_0\) \((x + y)^0 = 1\)
1 1 1 \({}_1 \mathrm{C}_0\), \({}_1 \mathrm{C}_1\) \((x + y)^1 = x + y\)
2 1 2 1 \({}_2 \mathrm{C}_0\), \({}_2 \mathrm{C}_1\), \({}_2 \mathrm{C}_2\) \((x + y)^2 = x^2 + 2xy + y^2\)
3 1 3 3 1 \({}_3 \mathrm{C}_0\) to \({}_3 \mathrm{C}_3\) \((x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\)
4 1 4 6 4 1 \({}_4 \mathrm{C}_0\) to \({}_4 \mathrm{C}_4\) \((x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\)

The numbers in Pascal’s Triangle are the same as the binomial coefficients:

\[\Large {}_n \mathrm{C}_r = \frac{n!}{r!(n-r)!}\]

The Binomial Theorem

The Binomial Theorem gives a formula for expanding \((x + y)^n\).

\[\Large (x + y)^n = \sum_{r=0}^{n} {}_n \mathrm{C}_r x^{n-r} y^r\]

Each term is formed by:

Example 1: Expanding Binomials

Expand \((x + 2)^3\).

Using Pascal’s Triangle row 3: 1 3 3 1

\[(x + 2)^3 = 1x^3 + 3x^2(2) + 3x(2^2) + 1(2^3)\]

\[= x^3 + 6x^2 + 12x + 8\]

The expanded form is \(x^3 + 6x^2 + 12x + 8\).

Binomial Probability Formula

The probability of exactly \(r\) successes in \(n\) trials is:

\[\Large P(r) = {}_n \mathrm{C}_r p^r q^{\,n-r}\]

Where:

Since there are only two outcomes,

\[\Large p + q = 1\]

Example 2: Finding Binomial Probabilities

A fair coin is tossed 4 times. What is the probability of getting exactly 3 heads?

\[n = 4, \quad r = 3, \quad p = \frac{1}{2}, \quad q = \frac{1}{2}\]

\[P(3) = {}_4 \mathrm{C}_3 \left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^{1}\]

\[= 4 \cdot \frac{1}{8} \cdot \frac{1}{2}\]

\[= 4 \cdot \frac{1}{16}\]

\[= \frac{4}{16} = \frac{1}{4}\]

The probability of exactly 3 heads is \(\Large \frac{1}{4}\).

Example 3: Problem-Solving Application

A basketball player makes 80% of free throws. If the player shoots 5 free throws, what is the probability the player makes exactly 4 shots?

\[n = 5, \quad r = 4, \quad p = 0.8, \quad q = 0.2\]

\[P(4) = {}_5 \mathrm{C}_4 (0.8)^4 (0.2)^1\]

\[= 5 (0.4096)(0.2)\]

\[= 5 (0.08192)\]

\[= 0.4096\]

The probability that the player makes exactly 4 shots is 0.4096, or 40.96%.

Normal Distributions

A random variable is a variable whose value is determined by chance.

A probability distribution describes how probabilities are assigned to each value of a random variable.

Discrete and Continuous Probability Distributions

A discrete probability distribution lists all possible values of a discrete random variable and their probabilities.

Example: The number of heads when tossing a coin 3 times (0, 1, 2, or 3).

A continuous probability distribution describes probabilities for a continuous random variable. The probability of any single value is 0, and probabilities are found over intervals.

Example: Heights of students in a school.

Normal Curve

A normal curve is a symmetric, bell-shaped curve that represents a normal distribution.

The curve is centered at the mean \(\mu\), and the spread is determined by the standard deviation \(\sigma\).

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean.

It is written as:

\[\Large X \sim N(\mu, \sigma)\]

To find probabilities, we convert values to z-scores.

\[\Large z = \frac{x - \mu}{\sigma}\]

Example: Finding Normal Probabilities

The scores on a standardized test are normally distributed with a mean of 70 and a standard deviation of 10. What is the probability that a student scores above 85?

Step 1: Find the z-score.

\[z = \frac{85 - 70}{10}\]

\[= \frac{15}{10}\]

\[= 1.5\]

Step 2: Use a z-table or calculator.

The area to the left of \(z = 1.5\) is approximately 0.9332.

Step 3: Find the probability above 85.

\[P(X > 85) = 1 - 0.9332\]

\[= 0.0668\]

The probability that a student scores above 85 is approximately 0.0668, or 6.68%.

Fitting to a Normal Distribution

Many real-life data sets are approximately normally distributed. When data follow a normal distribution, we can estimate probabilities and make predictions using the normal curve.

Example 1: Estimating Probabilities Using a Normal Curve

The heights of adult men in a city are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Estimate the probability that a randomly selected man is between 67 inches and 73 inches tall.

Step 1: Convert to z-scores.

For 67 inches:

\[z = \frac{67 - 70}{3} = \frac{-3}{3} = -1\]

For 73 inches:

\[z = \frac{73 - 70}{3} = \frac{3}{3} = 1\]

Step 2: Use the Empirical Rule.

About 68% of data in a normal distribution fall within 1 standard deviation of the mean.

Therefore,

\[P(67 < X < 73) \approx 0.68\]

The probability is approximately 68%.

Example 2: Using Standard Normal Values

The standard normal distribution is a normal distribution with:

\[\mu = 0 \quad \text{and} \quad \sigma = 1\]

A standard normal value is also called a z-score. It tells how many standard deviations a value is from the mean.

Example:

Test scores are normally distributed with a mean of 500 and a standard deviation of 100. What percent of students score below 650?

Step 1: Find the z-score.

\[z = \frac{650 - 500}{100}\]

\[= \frac{150}{100}\]

\[= 1.5\]

Step 2: Use a standard normal table.

The area to the left of \(z = 1.5\) is approximately 0.9332.

About 93.32% of students score below 650.

Example 3: Determining Whether Data May Be Normally Distributed

Not all data sets are normally distributed. To determine whether data may be approximately normal, consider the following:

Example:

A histogram of exam scores shows a symmetric, bell-shaped distribution centered near 75. The mean and median are both about 74.8.

Because the graph is symmetric and follows the Empirical Rule, the data may be approximately normally distributed.

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