MA121 Study Guide

2a. Apply simple principles of probability, and use common terminology of probability

• Define probability. Explain how to compute probability.
• Explain the concept of equally likely outcomes.
• Define a random variable. How is it different from variables you may have encountered in Algebra? Name the types of random variables.

Think about probability as the chance that an outcome or event will occur. The probability of something occurring is a number ranging between zero (zero percent or no chance) to one (100 percent or definitely).

We always express probability as a decimal for use in calculations. In other words, we write 55 percent as p = 0.55. If you have several events with equally likely outcomes then the number of possible outcomes is the denominator and the number of "successful" outcomes is the numerator.

For example, the probability of rolling greater than four on a six-sided die is 2/6, because rolling a five and six are the successes out of six outcomes. We cannot extend this to the sum of two dice. Totals of 2 through 12 make eleven possible events, but not all of them are equally likely. There is only one way {1,1} to roll a 2, but six ways {1,6}, {2,5}, {3,4}, {4,3}, {5,2}, {6,1} to roll a 7.

There are many variations of the probability formula for different situations and distributions, but we generally calculate probability as the number of "favorable" outcomes (that is, outcomes you are looking for) divided by the total number of possible outcomes.

A random variable differs from variables you have seen in algebra, because in those courses, the value of x is fixed and we solve its value by following certain steps.

For example, for the equation 2x = 4, x always equals two; it is just a matter of finding it. The study of probability introduces the concept of random variables: their value results from a probability experiment.

If x equals the number of times a coin lands on heads when you flip a coin five times, x will have a value from 0 to 5. We do not know the specific value until we have tossed the coin five times.

A random variable can be discrete (a specific set of possible values) or continuous (many or an infinite number of possible values). In our coin toss example above, x is a discrete random variable, since x can only have six possible values (0 to 5).

A continuous random variable might be a team's score during a basketball game. Just as we discussed above, there are too many possible values to make a frequency table for each possible point value.

Review this material in Remarks on the Concept of "Probability" and Random Variables.

2b. Calculate conditional probability, and determine whether two events are mutually exclusive and whether two events are independent

• Define an outcome and an event. How are the two related?
• What does it mean for two events to be dependent or independent?
• What does it mean for two events to be mutually exclusive?
• Define conditional probability.
• Define a compound event.

An outcome is the result of a single experiment. Examples include flipping a coin, measuring someone's height, or asking someone to name their favorite baseball team.

Two events are independent when the occurrence of one event or action does not depend on the occurrence of another.

• An example of independent events are: A = roll a five on a die, and B = flip heads on a coin. The two events do not have a causal relationship.
• An example of dependent events are: A = the temperature is below freezing, and B = it snows. In this case, event B is more likely to happen if A occurs than if it does not.

Compound events are combinations of events; we can calculate the probabilities for them as well. Common conjunctions are "AND", "OR", and "GIVEN" (conditional probability). In our above example, "Below freezing AND snowfall" is an example of the compound event (A and B).

Mutually-exclusive events (also called disjoint events) are events that cannot occur at the same time.

• An example of mutually-exclusive events are: A = roll greater than a four on a six-sided die, and B = roll less than three on a six-sided die. These events cannot occur at the same time.

Symbolically, we describe this as $P(A \cap B) = 0$

We pronounce the notation $P(A | B) = 0$ as "A given B" which means we are looking for the probability that A occurs given that B occurs, or has occurred. We call this situation conditional probability.

Review this material in Remarks on the Concept of "Probability" and Basic Concepts.

2c. Calculate probabilities using the addition rules and multiplication rules

• Define the general addition rule of probability. How does it relate to whether events are mutually exclusive, or not?
• Define the multiplication rule of probability. How does it relate to whether events are independent, or not?
• Define the special addition rule, general multiplication rule, and special multiplication rule.

We generally associate the general addition rule of probability with "or" compound events. We associate the multiplication rule with "and" compound events.

The probability p(A or B) = p(A) + p(B) holds if A and B are mutually exclusive (cannot both occur).

If A and B are not mutually exclusive, the special addition rule holds: p(A or B) = p(A) + p(B) − p(A & B).

The general multiplication rule is p(A & B) = p(A) × p(B) and holds if A and B are independent.

If they are dependent events, that's when conditional probability comes in and we use the special multiplication rule: p(A & B) = p(A) × p(B | A).

Review this material in:

• Probability with Playing Cards and Venn Diagrams
• Addition Rule for Probability
• Introduction to Probability

2d. Construct and interpret Venn diagrams

• Define a union or an intersection of events.
• Explain how to use Venn diagrams to illustrate outcomes and events.

The probability of a union of events $p(A\cup B)$ is the same as either A or B (or both) happening.

The probability of an intersection of events $p(A\cap B)$ is the same as saying that A and B both happen. We can represent these on a Venn diagram as events (circles) where outcomes are points in each circle. A union would be pictured as both circles shaded in, where an intersection would be represented as only the common area being shaded in.

Review this material in Probability with Playing Cards and Venn Diagrams and Addition Rule for Probability.

2e. Apply useful counting rules in the context of combinatorial probability

• Define and explain the difference between a combination and a permutation.

A combination and permutation refer to the number of possible ways that x out of a possible n outcomes can occur. The difference is that the order does not matter in a combination, but order does matter in a permutation.

For example, there are ten ways to choose x = 2 out of the first n = 5 letters of the alphabet: AB, AC, AD, AE, BC, BD, BE, CD, CE, DE. If order does not matter, ten combinations are possible. You could reverse the order of any of the letters and it would not matter.

There would be 20 permutations if AB and BA were considered different. Without listing them all here, you can intuitively see this because you have ten combinations and each combination can be in two different orders (AB or BA), so 10 × 2 = 20 permutations.

Review this material in Permutations and Combinations.

2f. Identify and use common discrete probability distribution functions

• Define a probability distribution. How does a probability distribution relate to the frequency tables we reviewed in Unit 1?
• What is the difference between a discrete and continuous probability distribution?

A probability distribution consists of each possible value (or interval of values) of a random variable, and the probability that the variable will take on that value.

Probability distributions have many implications in decision making. When you interpret data, you will need to know the probability distribution the value you are trying to estimate follows. They are related to the frequency tables in that the probabilities are equivalent to what we call the relative frequency distribution.

If you have a list of data, you can get the probabilities on the right side of the table by dividing the frequency by the total number of data points.

• For example, if the frequency of x = 3 is 7 in 20 die rolls, then the probability of rolling a three is 7/20 = 0.35, and so 0.35 would go across from value 3 in the table.

The difference between discrete and continuous probability distributions is analogous to the difference between discrete and continuous variables.

A discrete distribution (like a die roll) has a fixed, finite set of possible values for the random variable x, while a continuous distribution has many or infinitely many possible values for x. Like a frequency distribution, the values of x must be grouped into intervals. The same rules apply as those that apply to relative frequency histograms: all intervals must be equal width, non-overlapping, and all-inclusive.

Review this material in:

• Random Variables
• Probability Distributions for Discrete Random Variables
• Continuous Random Variables

2g. Calculate and interpret expected values

• What is the expected value of a distribution and how is it related to the mean of a set of data?

The expected value of a distribution is another name for the mean of the distribution.

This means that if you take a large set of numbers which follows the original distribution, the arithmetic mean (sum divided by n) of those numbers should roughly equal the expected value of the distribution.

We calculate the value of a distribution by multiplying each value of x by its probability (the median of the interval if it is continuous) and then summing up those numbers.

Review this material in Probability Distributions for Discrete Random Variables.

2h. Identify the binomial probability distribution, and apply it appropriately

• What is a binomial experiment?
• What is a binomial distribution? What characteristics do a set of events have to have to follow a binomial distribution?

A binomial experiment is an random experiment that has exactly two possible values (quantitative or qualitative) for x:

Flipping a coin: x = {heads, tails}

Answer to a true-false question: x = {true, false}

Free throw in basketball: x = {made, not made}

A binomial distribution is the distribution of the discrete random variable x, where x represents the number of successes out of n possible events. Note that we mean success in a generic sense: success may not be something positive. "Success" means the characteristic you are looking for or researching, regardless of whether you consider the outcome to be good or bad.

In our basketball free-throw example, if a player takes 10 free-throw shots, they will be successful zero to 10 times. Possible values for x include {0, 1, 2, 3, 4, 5, 6, 7, 8, 10}.

These values and their associated probabilities comprise a binomial probability distribution, which must have three criteria:

1. All experiments are binomial.
2. All experiments will succeed or fail independently of each other (that is, all free throws are independent events).
3. All experiments have an equal probability of success.

A binomial distribution is a family of distributions where each specific distribution consists of two parameters: n = the number of experiments, p = the probability of success per experiment.

For our basketball example, if the player hits their free throws 75% of the time, we would define this distribution as binomial with n = 10 and p = 0.75. There are an infinite number of possible binomial distributions, with each combination of n & p making a unique distribution. 

TIP: Any specific distribution within a family of distributions is defined by its parameter(s). For example, binomial distributions have parameters n & p.

We can calculate the expected value of this distribution by multiplying n and p, or n × p.

Review this material in The Binomial Distribution and Binomial Distribution.

2i. Identify the Poisson probability distribution, and apply it appropriately

• Define a Poisson distribution. When can you use it to estimate a binomial distribution? 
• Describe a second general use for Poisson distribution In addition to approximating a binomial distribution.
• A binomial distribution with n = 20 and p = 0.4 gives the same lambda (or mean), and the same Poisson distribution as a binomial distribution with n = 2000 and p = 0.004. How can the same Poisson distribution be "equivalent" to multiple binomial distributions?

The Poisson probability family of distributions (pronounced pwah-SOHN) has two main applications:

1. The Poisson distribution is related to the binomial distribution because you can use it to approximate the binomial distribution when n is a large value and p is a small value. Unlike the binomial distribution, the Poisson distribution is easier to calculate because it has only one parameter (represented by the Greek letter lambda λ) and the expected value. The calculation is n * p, just as for binomial distribution.
   
   
2. Statisticians refer to the Poisson distribution as the distribution of rare events. Suppose 1,000 cars drive on a section of road every day, and 1.2 of them get into an accident on average. Since the mean is so small compared to n, we can model this using a Poisson distribution with $\lambda =1.2$. We can use the formula to find the probability of x = 1 accident, x = 2 accidents, and so on.
   
   A minor difference between the Poisson distribution and binomial distribution is that the binomial distribution is a discrete distribution, where all possible values of x are between zero and n. The Poisson distribution is discrete in that x only has a fixed set of values: in theory it can have ANY whole number value from zero to infinity.
   
   For our car example, any probability for x greater than, let's say five, is so remote that it is effectively zero. Theoretically we could calculate the probability p(x = 150).
   
   How would you calculate Lambda? x = the expected number of occurrences during a fixed time period. So if a toll booth averages 45 cars per hour, and your fixed time period is 10 minutes, then the value of λ would be the expected number in 10 minutes, so 45 per 60 minutes would be 7.5. Note that although they are discrete, the expected value for the binomial and Poisson distributions do not have to be a whole number.

Finally, for each of the two distributions in the third question above lambda $\lambda =8$. However, you should see that finding the probability p(x = 7) gives the same answer for two different distributions. The Poisson distribution is a more accurate approximation for the second distribution than for the first. The Poisson distribution is a better predictor of the binomial distribution, the larger n gets (and the smaller p gets).

Review Poisson distribution in Poisson Distribution and Poisson Process 1 | Probability and Statistics | Khan Academy.

2j. Identify and use continuous probability density functions

• What is the difference, in general, between computing probabilities from discrete vs. continuous distributions? 
• Explain why one rule of continuous probability distributions is that there is zero probability that the random variable x equals any particular value.

In a discrete distribution, we can calculate the probability p(x = x) that x equals a particular value from a formula, depending on the distribution. We can also find probability in a range of values p(1 < x < 5) by computing p(x = 1) through p(x = 5) and adding the numbers. 

The difference between discrete and continuous distributions is that the probability p(x = x) that x equals a particular value is effectively zero. We can compute p(a < x < b) by finding the area under the density curve between x = a and x = b. It doesn't depend on the height of the graph.

This is why statisticians often call discrete distributions probability distribution functions and continuous distributions probability density functions. We use the term "density" because probabilities are based on the area between the two numbers, not the height of the graph. A probability density function, by definition, has an area of 1 (that is, it is unitless) under the entire curve.

We can explain this rationale in two ways. The most obvious reason is that p(x = x) is a single line which has no area if we calculate the probability that x is in a range of values by computing the area under the curve. The second reason is more conceptual. Since a continuous distribution has an infinite possible values of x, if we are talking about a uniform distribution between x = 0 and x = 1, an infinite number of numbers exist between those two values. Based on the definition of probability, since there are infinite possible outcomes, 1/∞ tends to 0.

Review this material in Continuous Random Variables.

2k. Identify the normal probability distribution, and apply it appropriately

• Describe the characteristics of a normal distribution. What sets it apart from any other bell-shaped distribution?
• What does it mean to be symmetric? Describe a uniform distribution.
• In general, how do we calculate probabilities based on a normal distribution?
• Define a standard normal distribution and what sets it apart from other normal distributions.

As we have discussed, probability distributions can be discrete or continuous. A continuous distribution can be symmetric if the density curve is symmetric around the median, which is also the mean.

A uniform distribution has a flat density curve. Another classic example of this is the discrete distribution where x = the sum of two six-sided die. There is only one way each to roll a 2 or a 12, but the median x = 7 has six possible ways: {1, 6}, {2, 5}, {3, 4}, {4, 3}, {5, 2}, and {6, 1}. The distribution has higher probabilities toward the mean/median and lower probabilities toward the edges.

A bell-shaped distribution has a bell-shaped density curve, like the dice distribution except continuous and graphically represented by a smooth curve.

Further in the hierarchy, we have the normal distributions which has all the characteristics above but with a few additional tell-tale characteristics, which we refer to as the empirical rule:

1. The probability of x having a value between 1 standard deviation under to 1 standard deviation over the mean is about 68 percent.
2. The probability of x being between −2 and +2 standard deviations is about 95 percent.
3. The probability of x being between −3 and +3 standard deviations is about 99.7 percent.

There is an important reason why we PLURALIZE "normal distributions" above. As we said earlier, we define distributions by their parameters, such as n & p for binomial distributions. A given combination of mean $\mu$ and standard deviation $\sigma$ makes a particular normal distribution.

Finally, the standard normal distribution is a normal distribution with mean = 0 and standard deviation = 1. We will need to obtain a standard normal distribution (often referred to as the Z distribution) to calculate probabilities involving all normal distributions.

To find the probability of x being between a and b in a normal distribution, we must take the following steps:

1. Convert the endpoint(s) into Z scores using the formula $Z=\frac{x-\mu}{\sigma}$
2. Use technology or a Z distribution table to look up the area to the left of b and the area to the left of a and subtract the two values.
3. For p(x < a) convert a into a Z score and find the area left of that value.
4. For p(x > b) convert b into a Z score, find the area left of that value and then subtract that number from 1.

In summary, the hierarchy is:

1. Probability distribution
2. Continuous probability distribution
3. Symmetric
4. Normal
5. Standard normal (though discrete distributions can also be symmetric).

Review this material in:

• Introduction to Normal Distributions
• Areas under Normal Distribution
• Areas of Tails of Distributions
• Probability Computations for General Normal Random Variables
• Introduction to the Normal Distribution

Unit 2 Vocabulary

• Addition (general and special) rules of probability
• Binomial distribution
• Binomial experiment
• Combination
• Compound event
• Conditional probability
• Continuous probability distribution and continuous random variable
• Dependent and independent events
• Discrete distribution and discrete random variable
• Empirical rule
• Event
• Expected value of a distribution
• Intersection
• Multiplication (general and special) rules of probability
• Mutually exclusive events
• Normal distribution
• Outcome
• Parameters
• Permutation
• Poisson distribution
• Probability density function
• Probability distribution
• Probability distribution function
• Relative frequency distribution
• Standard normal distribution
• Symmetric distribution
• Uniform distribution
• Union
• Venn diagram

3a. Apply the central limit theorem to approximate sampling distributions

• Define the sampling distribution of a mean.

Think again about the difference between sample and population statistics. When you take a sample from a population and measure its sample mean, then take a second sample and measure its mean, and keep going, the set of sample means will have its own distribution.

The central limit theorem states that if the original population had a normal distribution, OR the sample size is sufficiently large (n = 30 as a rule of thumb, but it can be a bit lower if the original population is CLOSE to normal), then the sample means themselves will be normally distributed, with a mean $\mu _{\overline{x}}\approx \mu _{x}$ and standard deviation of $\sigma _{\overline{x}}\approx \frac{\sigma_{x} }{\sqrt{n}}$. We often call this standard deviation the standard error.

Review this material in: 

• The Mean and Standard Deviation of the Sample Mean
• The Mean and Standard Deviation of the Sample Mean
• Sampling Distribution

3b. Describe the role of sampling distributions in inferential statistics

• How does the process of finding probabilities differ for a normal random variable versus the sampling distribution?

There is a subtle difference between the probabilities we are finding now and those we were finding at the end of Unit 2. When we first learned about probabilities from normal distributions, we were solving problems such as: If the population has mean and standard deviation of 10 and 2 respectively, find the probability that a random variable will have a value between 11 and 14.

The difference here is subtle, but important: If the population has mean and standard deviation of 10 and 2 respectively, find the probability that the mean of a sample of 10 taken from this population will have a value between 11 and 14.

Another way to think of this is that earlier you were working with sample size n = 1, so the above equation for standard error is the same as the population standard deviation, since you are dividing by one. So you still find the Z score, but rather than divide by σ, you must divide by $\sigma/{\sqrt{n}}$. The rest of the process is the same, taking those Z scores and using the tables or technology to find the appropriate areas under the curve.

Review this material in: 

• The Mean and Standard Deviation of the Sample Mean.
• The Sampling Distribution of the Sample Mean.
• Sampling Distribution

3c. Interpret and create graphs of a probability distribution for the mean of a discrete variable

• What is the mean of a discrete variable and how is it similar/different from the mean of a set of numbers?

The mean of a discrete random variable (also known as the expected value) is equal to the mean of a set of numbers that comes directly from that discrete distribution. Of course, this assumes no randomness, so the two numbers might be slightly different.

Take a very simple discrete distribution: p(X = 0) = 0.5, p(X = 1) = 0.2, p (x = 5) = 3.0. By the rules for mean of a discrete distribution (see resources below), the mean would be 0.5(0) + 0.2(1) + 0.3(5) = 1.7.

Now, let's take 10 numbers that perfectly represent this distribution: 0, 0, 0, 0, 0, 1, 1, 5, 5, 5. The sum is 17 and 17/10 = 1.7. So we get the same number from taking the mean of the distribution vs taking the mean of numbers from that distribution. Of course if we truly sample from the above distribution, because of randomness we're not going to get those 10 exact numbers, so taking the mean of 10 numbers drawn randomly from that distribution won't be exactly 1.7.

Review this material in Introduction to Sampling Distributions.

3d. Describe a sampling distribution in terms of repeated sampling

• What does it mean to take repeated samples from a population and what implication does this have?

Taking the means from repeated samples creates a sampling distribution. You can find the probability of $\overline{x}$ in roughly the same way you found the probability of x in Unit 2.

The formula for finding a Z score is different because you divide by standard error, rather than the standard deviation.

The reason this is important is because statistics is ultimately about making predictions about populations based on a sample (inferential statistics). A major step for being able to calculate margins of error is not only to observe properties of distributions, but to see how the resulting samples behave.

3e. Define and compute the mean and standard deviation of the sampling distribution of population proportion p

• What is the sampling distribution of a population proportion and how does it differ from the sampling distribution of a mean?
• How is the process for finding probabilities different? Can we still use the Z distribution?

Similar to the sampling distribution of the mean, you can take repeated samples from a population where p is the proportion having a certain characteristic, and the sample proportions will be normally distributed if the number sampled each time is sufficiently large.

A good rule of thumb is that given the parameters n = sample size and p = population proportion, we can use the normal distribution if np and n(1 – p) are both greater than 10 and p is not too close to 0 or 1. With a very small or large value for p, the distribution becomes very right or left skewed, and the process for finding areas based on the Z score is unreliable.

Mean $\mu _{\widehat{p}}\approx p$ and standard deviation $\sigma _{\widehat{p}}\approx \frac{\sqrt{p(1-p)}}{\sqrt{n}}$

Note that the formulas use the population proportion p rather than the sample proportion $\widehat{p}$. This becomes an important difference later when you have to make approximations of the population based on samples, and you of course do not have access to the population parameters.

Review this material in Sampling Distribution of a Proportion and The Sample Proportion.

3f. Identify or approximate a sampling distribution based on the properties of the population

• How do the properties of the population distribution affect the sampling distribution?

If you have a very large sample size, the short answer is that they do not. The sampling distribution will be bell-shaped and have a predictable mean and standard deviations.

This explanation does not hold for smaller standard deviations where the properties of the sampling distribution are far less predictable. In some cases we can come up with sampling distributions and properties through small-sample methods, but these examples are beyond the scope of this course.

3g. Compare and evaluate the sampling distributions of different sample sizes

• What effect does changing the sample size have on the sampling distribution of a mean?

Keep two things in mind here:

1. Unless the underlying population distribution is normal, the sample size must be sufficiently large for the sampling distribution to be normal.
   
2. As we stated above, the standard error (standard deviation of the sample means) becomes smaller with a larger sample. More specifically, it decreases by a factor of the square root of the sample size. For example, if you multiply the sample size by 4, you must divide the standard error by 2 (the square root of 4).

Review this material in Introduction to Sampling Distributions.

3h. Compare and evaluate the performance of different estimators based on their sampling distributions

• What is bias and how does it differ from accuracy? Can an estimator be biased yet accurate? Can it be inaccurate and unbiased?
• Why is the sample mean considered an unbiased estimator for the population mean?

No estimator is perfect. We use an estimator like a sample mean to give us the best possible estimate of a population mean. But because of sampling error the sample mean will always change for different samples.

What sampling error refers to is that because we do not have access to the entire population, our sample statistics will differ from the population parameter, and from each other. If you have a population of size 1,000 with a population mean of 50, you can take 5 samples of size 20, and because each time you have a different sample, you will get a different sample mean each time, such as {49, 51, 50, 48, 55}.

An unbiased estimator will sometimes estimate too high, sometimes too low, but you will get a good estimate in the long run if you average them. In other words, the sample mean is just as likely to overestimate the population mean by 2 as it is to underestimate by 2.

A biased estimator might be more likely to overestimate rather than underestimate, or vice versa. Accuracy differs from bias in that it refers to the amount that the statistics will be different than the parameter. It's important, but not as much as bias. Accuracy is going to be better, in general, when we have larger samples and there's less variance (or standard deviation) in the population.

Review this material in Characteristics of Estimators.

Unit 3 Vocabulary

• Bias
• Central Limit Theorem
• Estimator
• Random variable
• Sampling distribution
• Standard error
• Sampling error
• Unbiased estimator
• Z distribution

4a. Explain the central limit theorem, and use it to construct confidence intervals

• What role does the Central Limit Theorem have for estimation and sampling distributions?

We discussed the Central Limit Theorem in Unit 3. It is CRITICAL to note that if the distribution is non-normal, we MUST still have a large sample size. In other words, the Central Limit Theorem still applies. If it does not (non-normal distribution AND low sample size) then neither Z nor T will work.

The Central Limit Theorem still requires a normally distributed population or a sample size above 30. Using the T distribution instead of Z does not absolve us of these requirements.

Review the Central Limit Theorem in: 

• Confidence Intervals Introduction
• Confidence Interval for the Mean
• Large Sample Estimation of a Population Mean

4b. Compare t-distributions and normal distributions

1. What is the difference between a normal distribution and a T distribution?
2. Where would you need to use a T, instead of a normal (X), or the standard normal (Z) distributions?

Refer to learning objective 2k to review the progression from continuous distribution, to symmetric, to bell-shaped, to normal (X) and standard normal (Z).

Now, let's introduce the T distribution, which you can think of as a "brother" to the normal distribution in this hierarchy.

The Student's T distribution is similar to the normal distribution except that it is slightly shorter and flatter, with heavier tails than X or Z. In other words, the area to the right of two standard deviations in a normal distribution is 0.025. In a T distribution it will be larger. The T family of distributions is defined by a single parameter called the degrees of freedom which your book symbolizes simply as df although some texts will use other symbols like $\nu$. The degrees of freedom, when we are talking about the T distribution is equal to the sample size, minus one.

We use the T distribution in the same situations above as we use the Z, however we must use T instead when we are unsure of the shape of the population distribution, or we are using an estimate of the standard distribution (s instead of σ).

To find an area that is bound by the T distribution, we can use technology similar to what we use for the Z distribution. There is a T distribution table with one row each for most common values of T. The drawback is that we cannot use this to find the exact area, only the "T score" given a particular area.

Note that with a T distribution, as the sample size (thus the degrees of freedom) increases, it is nearly indistinguishable from the Z distribution. In fact, the Z distribution IS the T distribution with an "infinite" degrees of freedom! Using the T distribution gives us a larger margin of error to compensate for the fact that the exact standard deviation of the population is not known.

Review this material in T Distribution.

4c. Apply and interpret the central limit theorem for sample averages

• We saw earlier that the Central Limit Theorem holds if we have either a large sample size OR an underlying normal distribution. Does this affect whether we should use a Z or T distribution for estimation, or whether we can use either at all?

Whenever we perform an estimation, we must use a T distribution for a small sample size or an unknown value of the population standard distribution. The formula for using T distribution to find the margin of error and confidence intervals is the same as the one for Z, except we substitute σ with s. The difference is that instead of finding the cutoff value for the z score in the formula, we have to find the t score instead. This introduces an extra parameter. In addition to using the alpha value for the area right of the tail, we need to use n−1 (one less than the sample size) for the parameter "df". This will result in a larger margin of error because, as we stated earlier, T distributions have a heavier tail. 

Review this material in The Sampling Distribution of the Sample Mean.

4d. Calculate, describe, and interpret confidence intervals for population averages and one population proportions

• What does the confidence interval for a data sample tell you?
• Why was it necessary for you to learn about sampling distributions first?

We use inferential statistics to interpret samples and make conclusions about the population of data. The general method for making these interpretations is roughly the same, whether it is for population averages, proportions, averages of two different populations, standard deviations, or any other statistic.

First, we find a point estimate which is usually the sample mean or proportion. Then, given the level of confidence desired, the sample size, and the standard deviation of the population, we can find a margin of error and subtract or add it to the point estimate to get a confidence interval for the population parameter.

When we refer to level of confidence we mean the likelihood that our confidence interval contains the true population mean or proportion (or other parameter). Common confidence levels are 90%, 95%, and 99%. A higher confidence level gives a higher likelihood that the interval contains the population parameter. However the price to be paid is that naturally a higher confidence interval will be wider.

So for example, we could say there is a 95 percent probability that the population mean is between 10 ± 2.8, [7.2, 10.8]. If we want to be 99 percent confident, we have to increase the width of the interval, perhaps 10 ± 3.2. A 100 percent confidence interval is not possible since that would require an infinitely wide margin of error. So there is a give and take. Choosing a confidence level can be more art than science. Balance your desire for accuracy with the need to keep the margin of error low.

The reason you had to learn about sampling distributions first is because inferential statistics involve predicting the characteristics of a population based on a sample. In order to do so, we must first study how those samples behave.

Use the given formulas to calculate the margin of error for the population mean, given a large population (this is when you use the standard normal Z distribution). If the population is small, or it is large and the standard deviation of the population is unknown, you must use the Student's T.

The formulas are very similar, except for the distribution. You would use the inverse Z distribution or inverse T distribution with given degrees of freedom. You also have to plug in the Z or T score corresponding to ∝/2, where ∝ is equal to the tail area on one side. If you, for example, are trying to find a 90 percent confidence interval, that will leave 10 percent area on the tails, which you divide in half to come up with ∝/2 = .10/2 = 0.05.

Review this material in:

• Confidence Interval for the Mean
• Large Sample Estimation of a Population Mean
• Large Sample Estimation of a Population Proportion
• Confidence Interval Example
• Introduction to Confidence Intervals

4e. Interpret the student-t probability distribution as the sample size changes

• How is the student-t distribution related to sample size? Why does this not matter for normal or standard normal distributions?
• What happens to the student t distribution as the sample size increases? What distribution does it begin to resemble?

The student-t distribution, like the normal distribution, is a family of distributions defined by one or more parameters. A normal distribution is defined by its mean and standard deviation. The student-t distribution is defined by the number of degrees of freedom, which is equal to the sample size minus 1. 

The larger the sample size, the lower the margin of error, and the larger degrees of freedom, which combined will give you a smaller margin of error. The larger the degrees of freedom, the more the T distribution resembles the standard normal (Z) distribution. In fact, a Z distribution IS by definition the same as a T distribution with infinite degrees freedom!

Review this material in:

• Small Sample Estimation of a Population Mean
• Small Sample Estimation of a Population Mean
• Small Sample Size Confidence Intervals

Unit 4 Vocabulary 

• Central Limit Theorem
• Confidence interval
• Degrees of freedom
• Level of confidence
• Margin of error
• Normal distribution
• Point estimate
• Standard normal distribution
• Student's T distribution