# Overview of t-Tests: t-

Distribution-Based

Hypothesis Tests

A hypothesis test evaluates whether a sample is different enough from a population to establish that the sample probably did not come from that population. If a sample is different enough from a hypothesized population, then the population from which the sample came is different than the hypothesized population.

## Null Hypothesis

A hypothesis test is based upon a ** Null Hypothesis** which states that the sample did come from that population. A hypothesis test compares a sample statistic such as a sample mean to a population parameter such as the population’s mean. The amount of difference between the sample statistic and the population parameter determines whether the Null Hypothesis can be rejected or not.

The Null Hypothesis states that the population from which the sample came has the same mean or proportion as a hypothesized population. The Null Hypothesis is always an equality stating that the means or proportions of two populations are the same.

An example of a basic Null Hypothesis for a Hypothesis Test of Mean would be the following:

H_{0}: x_bar = Constant = 5

This Null Hypothesis would be used to state that the population from which the sample was taken has a mean equal to 5. The Constant (5) is the mean of the hypothesized population that the sample’s population is being compared to. The Null Hypothesis states that the sample’s population and the hypothesized population have the same means. The Alternative Hypothesis states that they are different.

An example of a basic Null Hypothesis for a Hypothesis Test of Proportion would be the following:

H_{0}: p_bar = Constant = 0.3

This Null Hypothesis would be used to state that the population from which the sample was taken has a proportion equal to 0.3. The Constant (0.3) is the proportion of the hypothesized population that the sample’s population is being compared to. The Null Hypothesis states that the sample’s population and the hypothesized population have the same proportions. The Alternative Hypothesis states that they are different.

### Null Hypothesis - Rejected or Not But Never Accepted

A hypothesis test has only two possible outcomes: the Null Hypothesis is either rejected or is not rejected. It is never correct to state that the Null Hypothesis was accepted. A hypothesis test only determines whether there is or is not enough evidence to reject the Null Hypothesis. The Null Hypothesis is rejected only when the hypothesis test result indicates a Level of Certainty that the Null Hypothesis is not valid at least equals the specified Level of Certainty.

If the required Level of Certainty for a hypothesis test is specified to be 95 percent, the Null Hypothesis will be rejected only if the test result indicates that there is at least a 95 percent probability that the Null Hypothesis is invalid. In all other cases, the Null Hypothesis would not be rejected. This is not equivalent to stating that the Null Hypothesis was accepted. The Null Hypothesis is never accepted; it can only be rejected or not rejected.

## Alternative Hypothesis

The Alternative Hypothesis is always in inequality stating that the means or proportions of two populations are not the same. The Alternative Hypothesis can be non-directional if it states that the means or proportions of two populations are merely not equal to each other. The Alternative Hypothesis is directional if it states that the mean or proportion of one of the populations is less than or greater than the mean of proportion of the other population.

An example of a non-directional Alternative Hypothesis for a Hypothesis test of Mean would be the following:

H_{1}: x_bar ≠ 5

This Alternative Hypothesis would be used to state that the population from which the sample was taken has a mean that is not equal to 5.

An example of a directional Alternative Hypothesis would be the following:

H_{1}: x_bar > 5

or

H_{1}: x_bar < 5

These Alternative Hypotheses would be used to state that the population from which the sample was taken has a mean that is either greater than or less than 5.

An example of a non-directional Alternative Hypothesis for a Hypothesis test of Proportion would be the following:

H_{1}: p_bar ≠ 0.3

This Alternative Hypothesis would be used to state that the population from which the sample was taken has a proportion that is not equal to 0.3.

An example of a directional Alternative Hypothesis would be the following:H_{1}: p_bar > 0.3

or

H_{1}: p_bar < 0.3

These Alternative Hypotheses would be used to state that the population from which the sample was taken has a proportion that is either greater than or less than 0.3.

## One-Tailed Test vs. Two-Tailed Test

The number of tails in a hypothesis test depends on whether the test is directional or not. The operator of the Alternative Hypothesis indicates whether or not the hypothesis test is directional. A non-directional operator (a “not equal” sign) in the Alternative Hypothesis indicates that the hypothesis test is a ** two-tailed test**. A directional operator (a “greater than” or “less than” sign) in the Alternative Hypothesis indicates that the hypothesis test is a

**.**

__one-tailed test__The Region of Rejection (the alpha region) for a **one-tailed test** is entirely contained in the one of the outer tails. A “greater than” operator in the Alternative Hypothesis indicates that the test is a one-tailed test in the right tail. A “less than” operator in the Alternative Hypothesis indicates that the test is a one-tailed test in the left tail. If α = 0.05, then one of the outer tails will contain the entire 5-percent Region of Rejection.

The Region of Rejection (the alpha region) for a two-tailed test is split between both outer tails. Each outer tail will contain half of the total Region of Rejection (alpha/2). If α = 0.05, then each outer tail will contain a 2.5-percent Region of Rejection if the test is a two-tailed tailed.

## Level of Certainty

Each hypothesis test has ** Level of Certainty** that is specified. The Null Hypothesis is rejected only when that Level of Certainty has been reached that the sample did not come from the population. A commonly specified Level of Certainty is 95 percent. The Null Hypothesis would only be rejected in this case if the sample statistic was different enough from the population parameter that at least 95 percent certainty was achieved that the sample did not come from that population.

## Level of Significance (Alpha)

The Level of Certainty for a hypothesis test is often indicated with a different term called the ** Level of Significance** also known as α (

**). The relationship between the Level of Certainty and α is the following:**

__alpha__

## α = 1 – Level of Certainty

An alpha that is set to 0.05 indicates that a hypothesis test requires a Level of Certainty of 95 percent that the sample came from a different population to be reached before the Null Hypothesis is rejected.

## Region of Acceptance

A Hypothesis Test of Mean or Proportion can be performed if the Test Statistic is distributed according to the normal distribution or the t distribution. The Test Statistic is derived directly from the sample statistic such as the sample mean. If the Test Statistic is distributed according to the normal or t distribution, then the sample statistic is also distributed according to normal or t distribution. This will be discussed is greater detail shortly.

A Hypothesis Test of Mean or Proportion can be understood much more intuitively by mapping the sample statistic (the sample mean or proportion) to its own unique normal or t distribution. The sample statistic is the distributed variable whose distribution is mapped according its own unique normal or t distribution

The Region of Acceptance is the percentage of area under this normal or t distribution curve that equals the test’s specified Level of Certainty. If the hypothesis test requires 95 percent in order to reject the Null Hypothesis, the Region of Acceptance will include 95 percent of the total area under the distributed variable’s mapped normal or t distribution curve.

If the observed value of the sample statistic (the observed mean or proportion of the single sample taken) falls inside of the Region of Acceptance, the Null Hypothesis is not rejected. If the observed value of the sample statistic falls outside of the Region of Acceptance (into the Region of Rejection), the Null Hypothesis is rejected.

## Region of Rejection

The Region of Rejection is the percentage of area under this normal or t distribution curve that equals the test’s specified Level of Significance (alpha). It is important to remember the following relationship:

Level of Significance (alpha) = 1 – Level of Certainty.

If the required Level of Certainty to reject the Null Hypothesis is 95 percent, then the following are true:

Level of Certainty = 0.95

Level of Significance (alpha) = 0.05

The Region of Acceptance includes 95 percent of the total area under the normal or t distribution curve that maps the distributed variable, which is the sample statistic (the sample mean or proportion).

The Region of Rejection includes 5 percent of the total area under the normal or t distribution curve that maps the distributed variable, which is the sample statistic (the sample mean or proportion). The 5-percent alpha region is entirely contained in one of the tails if the test is a one-tailed test. The 5-percent alpha region is split between both of the outer tails if the test is a one-tailed test.

If the observed value of the sample statistic (the observed mean or proportion of the single sample taken) falls inside of the Region of Rejection (outside the Region of Acceptance), the Null Hypothesis is rejected. If the observed value of the sample statistic falls inside of the Region of Acceptance, the Null Hypothesis is not rejected.

## Critical Value(s)

Each hypothesis test has one or two Critical Values. A Critical Value is the location of boundary between the Region of Acceptance and the Region of Rejection. A one-tailed test has one critical value because the Region of rejection is entirely contained in one of the outer tails. A two-tailed test has two Critical Values because the Region of Rejection is split between the two outer tails.

The Null Hypothesis is rejected if the sample statistic (the observed sample mean or proportion) is farther from the curve’s mean than the Critical Value on that side. If the sample statistic is farther from the curve’s mean than the Critical value on that side, the sample statistic lies in the Region of Rejection. If the sample statistic is closer to the curve’s mean than the Critical value on that side, the sample statistic lies in the Region of Acceptance.

## Test Statistic

Each hypothesis test calculates a ** Test Statistic**. The Test Statistic is the amount of difference between the observed sample statistic (the observed sample mean or proportion) and the hypothesized population parameter (the Constant on the right side of the Null Hypothesis) which will be located at the curve’s mean.

This difference is expressed in units of ** Standard Errors**. The Test Statistic is the number of Standard Errors that are between the observed sample statistic and the hypothesized population parameter. The Null Hypothesis is rejected if that number of Standard Errors specified by the Test Statistic) is larger than a critical number of Standard Errors. The critical number of Standard Errors is determined by the required Level of Certainty

### Test Statistic – z Score or t Value

The ** Test Statistic** is either the

**or the**

__z Score__**depending on whether a z Test or t-Test is being performed. This will be discussed in greater detail shortly.**

__t Value__##### Critical t Value or Critical z Value

Each hypothesis test calculates ** Critical t or z Values**. A Critical t Value is calculated for a t-Test and a Critical z Value is calculated for a z Test. A Critical t or z Value is the amount of difference expressed in Standard Errors between the boundary of the Region of Rejection (the Critical Value) and hypothesized population parameter (the Constant on the right side of the Null Hypothesis) which will be located at the curve’s mean.

A one-tailed test has only one Critical t or z Value because the Region of Rejection is entirely contained in one outer tail A two-tailed test has two Critical z or t Values because the Region of Rejection is split between the two outer tails.The Test Statistic (the t Value or z Score) are compared with the Critical t or z Value on that side of the mean.

If the Test Statistic is farther from the standardized mean of zero than the Critical t or z Value on that side, the Null Hypothesis is rejected. The Test Statistic is the number of Standard Errors that the sample statistic is from the curve’s mean.

The Critical t or z Value on the same side is the number of Standard Errors that the Critical Value (the boundary of the Region of Rejection) is from the mean. If the Test Statistic is farther from the standardized mean of zero than the Critical t or z value, the sample statistic lies in the Region of Rejection.

### p Value

Each hypothesis test calculates a p Value. The p Value is the area under the curve that is beyond the sample statistic (the observed sample mean or proportion). The p Value is the probability that a sample of size n with the observed sample mean or proportion could have occurred if the Null Hypothesis were true.

If, for example, the p Value of a Hypothesis Test of Mean or Proportion were calculated to be 0.0212, that would indicated that there is only a 2.12 percent chance that a sample of size n would have the observed sample mean or proportion if the Null Hypothesis were true. The Null Hypothesis states that the population from which the sample came has the same mean as the hypothesized population. This mean is the Constant on the right side of the Null Hypothesis.

The p Value is compared to alpha for a one-tailed test and to alpha/2 for a two-tailed test. The Null Hypothesis is rejected if p is smaller than α for a one-tailed test or if p is smaller than α/2 for a two-tailed test. If the p Value is smaller than α for a one-tailed test or smaller than α/2 for a two-tailed test, the sample statistic is in the Region of Rejection.

Calculations of the Critical t Value(s) and the p Value are as follows:

__Critical t Value For 1-Tailed Test in Right Tail:__

Excel 2010 and beyond

Critical t Value = T.INV(1-α,df)

Prior to Excel 2010

Critical t Value = T.INV(2*α,df)

__Critical t Value For 1-Tailed Test in Left Tail:__

Excel 2010 and beyond

Critical t Value = T.INV(α,df)

Prior to Excel 2010

Critical t Value = -TINV(2*α,df)

Note that the negative sign has to be manually inserted into this pre-2010 formula to calculate the Critical t Value in the left tail for a one-tailed test.

__Critical t Values For a 2-Tailed Test:__

Excel 2010 and beyond

Critical t Values = ±T.INV(1-α/2,df)

Prior to Excel 2010

Critical t Values = ±TINV(α,df)p Value

The p Value is calculated using the same formulas whether the test is a one-tailed test or a two-tailed test.

Excel 2010 and beyond

p Value = T.DIST.RT(ABS(t Value), df)

Prior to Excel 2010

p Value = TDIST(ABS(t Value), df)

## 3 Equivalent Reasons To Reject

Null Hypothesis

The Null Hypothesis of a Hypothesis Test of Mean or Proportion is rejected if any of the following equivalent conditions are shown to exist:

**1) Sample Statistic Beyond Critical Value**

The sample statistic (the observed sample mean or proportion) would therefore lie in the Region of Rejection because the Critical Value is the boundary of the Region of Rejection.

**2) Test Statistic Beyond Critical t or z Value**

The Test Statistic (the t Value of z Score) is the number of Standard Errors that the sample statistic is from the curve’s mean. The Critical t or z Value is the number of Standard Errors that the boundary of the Region of Rejection is from the curve’s mean. If the Test Statistic is farther from farther from the standardized mean of 0 than the Critical t or z Value, the sample statistic lies in the Region of Rejection.

**3) p Value Smaller Than α (1-Tailed) or α/2 (2-Tailed)**

The p Value is the curve area beyond the sample statistic. α and α/2 equal the curve areas contained by the Region of Rejection on that side for a one-tailed test and a two-tailed test respectively. If the p value is smaller than α for a one-tailed test or α/2 for a two-tailed test, the sample statistic lies in the Region of Rejection.

## Independent vs. Dependent Samples

A sample that is independent of a second sample has data values that are not influenced by any of the data values within the second sample. Dependent samples are often referred to as paired data. Paired data are data pairs in which one of the values of each pair has an influence on the other value of the data pair. An example of a paired data sample would be a set of before-and-after test scores from the same set of people.

## Pooled vs. Unpooled Tests

A two-independent-sample Hypothesis Test of Mean can be pooled or unpooled. A pooled test can be performed if the variances of both independent samples are similar. This is a pooled test because a single pooled standard deviation replaces both sample standard deviations in the calculation of the Standard Error. An unpooled test must be performed when the variances of the two independent samples are not similar.

## Type I and Type II Errors

A ** Type I Error** is a false positive and a

**is a false negative. A false positive occurs when a test incorrectly detects of a significant difference when one does not exist. A false negative occurs when a test incorrectly fails to detect a significant different when one exists.**

__Type II Error__α (the specified Level of Significance) = a test’s probability of a making a Type I Error.

β = a test’s probability of a making a Type II Error.

## Power of a Test

The Power of a test indicates the test’s sensitivity. The Power of a test is the probability that the test will detect a significant difference if one exists. The Power of a test is the probability of not making a Type II Error, which is failing to detect a difference when one exists. A test’s Power is therefore expressed by the following formula:

Power = 1 – β

## Effect Size

Effect size of Hypotheses Tests of Mean is usually expressed in measures of Cohen’s d. Cohen’s d is a standardized way of quantifying the size of the difference between the two groups. This standardization of the size of the difference (the effect size) enables classification of that difference in relative terms of “large,” “medium,” and “small.” A large effect would be a difference between two groups that is easily noticeable with the measuring equipment available. A small effect would be a difference between two groups that is not easily noticed.

## Nonparametric Alternatives for t-

Tests in Excel

Nonparametric tests are sometimes substituted for t-Tests because normality requirements cannot be met. A t-Test is a Hypothesis Test of Mean that can be performed if the sample statistic (and therefore the Test Statistic) is distributed according to the t distribution under the Null Hypothesis. The sample statistic (the sample mean) is distributed according to the t distribution if any of the following three conditions exist:

1) Sample size is large (n > 30). The sample taken for the hypothesis test must have at least 30 data observations.

2) The population from which the sample was taken is verified to be normally distributed.

3) The sample is verified to be normally distributed.

If none of these conditions can be met or confirmed, a nonparametric test can often be substituted for a t-Test. A nonparametric test does not have normality requirements that a parametric test such as a t-Test does.

## Hypothesis Test of Mean vs.

Proportion

Hypothesis Test covered in this section will either be ** Hypothesis Tests of Mean** or

**. A data point of a sample taken for a Hypothesis Test of Mean can have a range of values. A data point of a sample taken for a Hypothesis Test of Proportion is binary; it can take only one of two values.**

__Hypothesis Test of Proportion__

### Hypothesis Tests of Mean – Overview

A Hypothesis Test of Mean compares an observed sample mean with a hypothesized population mean to determine if the sample was taken from the same population. An example would be to compare a sample of monthly sales of stores in one region to the national average to determine if mean sales from the region (the population from which the sample was taken) is different than the national average (the hypothesized population parameter). As stated, a sample taken for a Hypothesis Test of Mean can have a range of values. In this case, the sales of a sample sampled store can fall within a wide range of values.

### Hypothesis Tests of Proportion – Overview

A Hypothesis Test of Proportion compares an observed sample proportion with a hypothesized population proportion to determine if the sample was taken from the same population. An example would be to compare the proportion of defective units from a sample taken from one production line to the proportion of defective units from all production lines to determine if the proportion defective from the one production line (the population from which the sample was taken) is different than from the proportion defective of all production lines (the hypothesized population parameter). As stated, a sample taken for a Hypothesis Test of Proportion can only have one of two values. In this case, a sampled unit from a production line is either defective or it is not.

Hypothesis Test of Proportion are covered in detail in a separate section in this manual. They are also summarized at the end of the binomial distribution section.

## Hypothesis Tests of Mean

Hypothesis Tests of Mean require that the Test Statistic is distributed either according to the normal distribution or to the t distribution. The Test Statistic in a Hypothesis Test of Mean is derived directly from the sample mean and therefore has the same distribution as the sample mean.

## t-Test vs. z Test

A Hypothesis Test of Mean will either be performed as a z Test or as a t-Test. When the sample mean and therefore the Test Statistic are distributed according to the normal distribution, the hypothesis test is called a ** z Test** and the Test Statistic is called the

**. When the sample mean and therefore the Test Statistic is distributed according to the t distribution, the hypothesis test is called a**

__z Score__**and the Test Statistic is called the**

__t-Test__**. The Test Statistic is the number of Standard Errors that the observed sample mean is from the hypothesized population mean.**

__t Value__t-Tests are covered in detail is a separate section in this manual. They are also summarized at the end of this t distribution section.

z Tests are covered in detail is a separate section in this manual. They are also summarized at the end of the normal distribution section.

### Means of Large Samples Are Normally Distributed

According to the Central Limit Theorem, the means of large samples will be normally distributed no matter how the population from which the samples came is distributed. This is true as long as the samples are random and the sample size, n, is large (n > 30). n equals the number of data observations that each sample contains.

If the single sample taken for a Hypothesis Test of Mean is large (n > 30), then the means of a number of similar samples taken from the same population would be normally distributed as per the Central Limit Theorem. This is true no matter how the population or the single sample are distributed.

If the single sample taken for a Hypothesis Test of Mean is small (n < 30), then the means of a number of similar samples taken from the same population would be normally distributed only if the population was proven to be normally distributed or if the sample was proven to be normally distributed.

## Requirements of a z-Test

A z Test can be performed only if the sample mean (and therefore the Test Statistic, which is derived from the sample mean) is normally distributed. The sample mean and therefore the Test Statistic are normally distributed only when the following two conditions are __both__ met:

1) The size of the single sample taken is large (n > 30). The Central Limit Theorem states that means of large samples will be normally distributed. When the size of the single sample is small (n < 30), only a t-Test can be performed.

2) The population standard deviation, σ (sigma), is known.

## Requirements of a t-Test

A ** t-Test** can be performed only if the sample mean (and therefore the Test Statistic, which is derived from the sample mean) is distributed according to the t distribution. The sample mean and therefore the Test Statistic are distributed according to the t distribution when

__both__of these conditions are met:

1) The sample standard deviation, s, is known.

2) Either the sample or the population has been verified for normality.

A t-Test can be performed when the single sample is large (n > 30) but is the only option when the size of the single sample is small (n < 30). A z Test can only be performed when the size of the single sample is large (n > 30) and the population standard deviation is known.

As mentioned, a Hypothesis Test of Mean requires that the sample mean and therefore the Test Statistic is distributed either according to the normal distribution or to the t distribution. The sample mean and the Test Statistic are distributed variables that can be graphed according to the normal or t distribution.

The Test Statistic, which represents the number of Standard Errors that the sample mean is from the hypothesized population mean, could be graphed on a standard normal distribution curve or a standardized t distribution curve. Both these two distribution curves have their means at zero and the length of one Standard Error is set to equal 1.

## Basic Steps of a Hypothesis Test of

Mean

The major steps of the simple Hypothesis Test of Mean, a one-sample t-Test, are described as follows:

1) A sample of data is taken. The sample statistic which is the sample mean is calculated.

2) A Null Hypothesis is created stating the population from which the sample was taken has the same proportion as a hypothesized population proportion. An Alternative Hypothesis is constructed stating that sample population’s proportion is not equal to, greater than, or less than the hypothesized population proportion depending on the wording of the problem.

3) The sample proportion is mapped to a normal curve that has a mean equal to the hypothesized population proportion and a Standard Error calculated based upon a formula specific to the type of Hypothesis Test of Proportion.

4) The Critical Values are calculated and the Regions of Acceptance and Rejection are mapped on the normal graph that maps the distributed variable. The Critical Values represent the boundaries between the Region of Acceptance and Region of Rejection.

5) Critical t Values, the Test Statistic (the t Value) and p Value are then calculated.

6) The Null Hypothesis is rejected if any of the following three equivalent conditions are shown to exist:

a) The observed sample mean, x_bar, is beyond the Critical Value.

b) The t Value (the Test Statistic) is farther from zero than the Critical t Value.

c) The p Value is smaller than α for a one-tailed test or α/2 for a two-tailed test.

The following graph represents the final result of a typical one-sample, two-tailed t-Test. In this case the Null Hypothesis was rejected. This result is represented as follows:

*(Click On Image To See a Larger Version)*

This t-Test was a two-tailed test as evidenced by the yellow Region of Rejection split between the both outer tails. In this t-Test the alpha was set to 0.05. This 5-percent Region of Rejection is split between the two tails so that each tail contains a 2.5 percent Region of Rejection.

The mean of this non-standardized t-distribution curve is 186,000. This indicates that the Null Hypothesis is as follows:

H_{0}: x_bar = 186,000

Since this is a two-tailed t-Test, the Alternative Hypothesis is as follows:

H_{1}: x_bar ≠ 186,000

This one-sample t-Test is evaluating whether the population from which the sample was taken has a population mean that is not equal to 186,000. This is a non-directional t-Test and is therefore two-tailed.

The sample statistic is the observed sample mean of this single sample taken for this test. This observed sample mean is calculated to be 200,000.

The boundaries of the Region of Rejection occur at 172,083 and 199,916. Everything beyond these two point is in the Region of Rejection. These two Critical Values are 1.95 Standard Errors from the standardized mean of 0. This indicates that the Critical t Values are ±1.95.

The graph shows that the sample statistic (the sample mean of 200,000) falls beyond the right Critical value of 199,916 and is therefore in the Region of Rejection.

The sample statistic is 2.105 Standard Errors from the standardized mean of 0. This is further from the standardized mean of 0 than the right Critical t value which is 1.95.

The curve area beyond the sample statistic consists of 2.4 percent of the area under the curve. This is smaller than α/2 which is 2.5 percent of the total curve area because alpha was set to 0.05.

As the graph shows, all three equivalent conditions have been met to reject the Null Hypothesis. It can be stated with at least 95 percent certainty that the mean of the population from which the sample was taken does not equal the hypothesized population mean of 186,000.

## Uses of Hypothesis Tests of Mean

**1) Comparing the mean of a sample taken from one population with the another population’s mean to determine if both populations have the different means.** An example of this would be to compare the mean monthly sales of a sample of retail stores from one region to the national mean monthly store sales to determine if the mean monthly sales of all stores in the one region are different than the national mean.

**2) Comparing the mean of a sample taken from one population to a fixed number to determine if that population’s mean is different than the fixed number.** An example of this might be to compare the mean product measurement taken a sample of a number of units of a product to the company’s claims about that product specification to determine if the actual mean measurement of all units of that company’s product is different than what the company claims it is.

**3) Comparing the mean of a sample from one population with the mean of a sample from another population to determine if the two populations have different means.** An example of this would be to compare the mean of a sample of daily production totals from one crew with the mean of a sample of daily production totals from another crew to determine if the two crews have different mean daily production totals.

**4) Comparing successive measurement pairs taken on the same group of objects to determine if anything has changed between measurements.** An example of this would be to evaluate whether there is mean difference in before-and-after tests scores of a small sample of the same people to determine if a training program made a difference to all of the people who underwent it.

**5) Comparing the same measurements taken on pairs of related objects**

__.__An example of this would be to evaluate whether there is mean difference in the incomes of husbands and wives in a sample of married couples to determine if there is a mean difference in the incomes of husbands and wives in all married couples.

It is important to note that a hypothesis test is used to determine if two populations are different. The outcome of hypothesis test is to either reject or fail to reject the Null Hypothesis. It would be incorrect to state that a hypothesis test is used to determine if two populations are the same.

## Types of Hypothesis Tests of Mean

Hypothesis Tests of Mean are either t-Tests or z-Tests.

__The 4 types of t-Tests performed in later articles of this blog are the following:__

One-sample t-Test

Two-Independent-Sample, Pooled t-Test

Two-Independent-Sample, Unpooled t-Test

Two-Dependent-Sample (Paired) t-Test

__The 3 types of z-Test performed in later articles of this blog are the following:__

One-sample z-Test

Two-independent-Sample, Unpooled z-Test

Two-Dependent-Sample (Paired) z-Test

**Each of the above z-Tests and t-Tests will be performed in detail in later articles in this blog.**

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