If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

It is the **most powerful prayer**.
A **pure heart**, a **clean mind**, and a **clear conscience** is necessary for it.

God is our refuge and our strength, our ever-present help in distress.

- Psalm 46:1

The

- Samuel Chukwuemeka**Joy** of a **Teacher** is the **Success** of his **Students.**

I greet you this day,

__First:__ Review the Notes.

__Second:__ View the Videos.

__Third:__ Solve the questions/solved examples.

__Fourth:__ Check your solutions with my **thoroughly-explained solved examples.**

__Fifth:__ Check your answers with the calculators.

I wrote the codes for the calculators using JavaScript, a client-side scripting language and AJAX, a JavaScript library. Please use the latest Internet browsers. The calculators should work.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me. If you are my student, please do not contact me here. Contact me via the school's system.

**Samuel Dominic Chukwuemeka** (SamDom For Peace)
B.Eng., A.A.T, M.Ed., M.S

Students will:

(1.) Discuss the meaning of hypothesis testing.

(2.) Explain the meaning of null and alternative hypothesis.

(3.) State the null and alternative hypothesis from a given claim.

(4.) State the type of hypothesis test.

(5.) State the population parameter being tested.

(6.) Explain the meaning of Type I and Type II errors.

(7.) Discuss social injustice related to Type I and Type II errors. (Relate Statistics to Criminology).

(8.) Identify the Type I and Type II errors from a claim.

(9.) Identify the Type I and Type II errors in social cases. (Relate Statistics to Criminology).

(10.) Disucss the three methods used in hypothesis testing.

(11.) Test the hypothesis for a claim using the Critical Method (Classical Approach).

(12.) Test the hypothesis for a claim using the Probability Value Method (P-value Approach).

(13.) Test the hypothesis for a claim using the Confidence Interval Method.

(14.) Write the decision of the hypothesis test based on the methods used.

(15.) Write the conclusion of the hypothesis test based on the decision.

(16.) Interpret the conclusion.

(17.) Discuss the power of a hypothesis test.

*
Recall:
Ask students to list the steps in the Scientific Method
Is Forming a Hypothesis one of the methods?
Is Testing the Hypothesis also one of the methods?
If you answered "yes", then you are right!
*

Bring it to **Logic:** A **hypothesis** is defined as the **premise of a conditional sattement.**

Bring it to **Science:** A **hypothesis** is **an unproven theory about a Scientific problem.**

Bring it to **Statistics:** A **hypothesis** is **a claim about a population parameter.**

We are testing a **claim** about a **population parameter** using a **sample statistic.**

We are testing a **claim** about a **population proportion** using a **sample proportion.**

We are testing a **claim** about a **population mean** using a **sample mean.**

We are testing a **claim** about a **population variance** using a **sample variance.**

We are testing a **claim** about a **population standard deviation** using a **sample standard deviation.**

A **hypothesis test** is a **procedure for testing a claim about a population parameter.**

The **null hypothesis** is the **statement** that shows that the value of the **population parameter is equal to some claimed value.**

It is denoted by $H_0$

We test the null hypothesis assuming it to be true.

We make a decision based on the null hypothesis.

We either reject the null the hypothesis; or we do not reject the null hypothesis (or fail to reject the null hypothesis)
based on the result of the methods we used for the test.

"Fail to reject the null hypothesis" is the same as "Do not reject the null hypothesis".

This does not mean that we accept the null hypothesis. It just means that we do not reject it.

*Explain this concept with examples.*

Then, we make a conclusion based on our decision.

The **alternative hypothesis** is the **statement** that shows that the value of the **population parameter is different from the claimed value.**

It is denoted by $H_1$ or $H_A$ or $H_a$

The **population parameter being different from the claimed value** means that it could be **less than the claimed value**; or
**greater than the claimed value**; or **not equal to the claimed value**.

If the **population parameter is less than the claimed value,** the **hypothesis test is a left-tailed test.**

In **left-tailed tests,** the **critical region is in the extreme left region (left tail).**

$population\:\:parameter \lt claimed\:\:value \implies left-tailed\:\:test$

If the **population parameter is greater than the claimed value,** the **hypothesis test is a right-tailed test.**

In **right-tailed tests,** the **critical region is in the extreme right region (right tail).**

$population\:\:parameter \gt claimed\:\:value \implies right-tailed\:\:test$

Left-tailed tests and Right-tailed tests are one-tailed tests.

If the **population parameter is not equal to the claimed value,** the **hypothesis test is a two-tailed test.**

In **two-tailed tests,** the **critical region is in the two extreme regions (two tails: left tail and right tail).**

$population\:\:parameter \ne claimed\:\:value \implies two-tailed\:\:test$

The **test statistic** is a **used in making a decision about the null hypothesis.**

It is found by converting the sample statistic to a score with the assumption that the null hypothesis is true.

There are three methods used in hypothesis testing.

Two of those methods will always lead to the same decision, and ultimaltely the same conclusion.

Those are the two main methods.

The other method may be used **only when those two main methods are used.**

The two main methods are:

(1.) **Critical Value Method** or **Classical Approach** or **Traditional Method**

**
If the test statistic is in the critical region, reject the null hypothesis.
If the test statistic is not in the critical region, do not reject the null hypothesis.
**

The

For a

For a

For a

(2.)

If the P-value is greater than the level of significance, do not reject the null hypothesis.

The other method is:

(3.)

Sometimes, we use the Confidence Interval method to test the hypothesis.

We typically use this method for:

If the confidence interval contains the value of the population parameter stated in the null hypothesis, do not reject the null hypothesis.

For

For

NOTE:

(I.) The

(II.) For

(III.) For

What if we formed our hypothesis correctly, applied the correct methods to test it, performed our calculation correctly, and still made a wrong decision which leads to a wrong conclusion?

Is it possible?

YES. It is possible.

Why?

We are humans. We are not GOD.

Only GOD does not make mistakes.

Humans are known for making mistakes.

This leads us to....

There are twi main types of errors we can make when testing hypothesis.

They are:

**Type I Error** (Rejecting a true null hypothesis)

This is the error made when we reject the null hypothesis when it is true.

*
Compare it to a False Positive scenario in Probability (Questions 50 and 51). Explain.
It is also similar to convicting an innocent man
*

This is the error made when we fail to reject the null hypothesis when it is false.

It is also similar to

Both errors are bad.

However, which one do you think is worse?

Type I error OR Type II error?

Would you convict an innocent man? OR Would you acquit a guilty man?

Truth |
|||

$H_0$ is true | $H_0$ is false | ||

Decision |
Reject $H_0$ | Type I error | Correct decision |

Do not reject $H_0$ | Correct decision | Type II error |

Compare this to: (You may use this to remember the main table)

Truth |
|||

Innocent | Guilty | ||

Verdict |
Convict | Type I error | Correct decision |

Acquit | Correct decision | Type II error |

$\alpha$ is the probability of making a Type I error.

$\beta$ is the probability of making a Type II error.

Type I and Type II errors are inversely related.

As $\alpha$ increases, $\beta$ decreases.

As $\alpha$ decreases, $\beta$ increases.

*Ask students to define the level of significance (when we covered Inferential Statistics)*

We are going to give another definition of the level of significance (as it concerns Hypothesis Testing).

The **level of significance** is the **probability of making the mistake of rejecting the null hypothesis when it is true.**

This implies that: The **level of significance** is the **probability of making a Type I error.**

It is denoted by $\alpha$.

NOTE: If $\alpha$ is not given, use $\alpha = 5\%$

The **power of a hypothesis test** is the **probability of rejecting a null hypothesis when it is false.**

This implies that: The **power of a hypothesis test** is the **probability of making the correct decision by avoiding making a Type II error.**

The power of a hypothesis test depends on the significance level, the sample size, and how wrong the null hypothesis is.

(1.) The samples are simple random samples.

(2.) There is a fixed number of trials.

(3.) The trials are independent.

(4.) Each trial results in either a success or a failure.

(5.) The probability of success or failure in any trial is the same as the probability of success or failure in all the trials.

(6.) There are at least ten successes and ten failures.

$np \ge 10$ AND $nq \ge 10$

(7.) The sample size is no more than five percent (at most five percent) of the population size.

$n \le 5\%N$ or $n \le 0.05N$

NOTE: If the population proportion is not given, use $50\%$

If $p$ is not given, use $p = 50\%$ or $p = 0.5$

(1.) The sample proportions are from two simple random independent samples.

Independent samples means that the sample values selected from one population proportion are not related to, or
somehow natuarlly paired or matched with the sample values from the other population.

(2.) There are at least five successes and five failures for each of the two samples.

$n\hat{p} \ge 5$ AND $n\hat{q} \ge 5$ for each of the two samples.

The two main **Hypothesis Test Methods** uses the **pooled sample proportion.**

This means that the data is pooled (sort of like "pooling resources together") to obtain the proportion of successes
in both samples combined.

The **Confidence Interval Method** uses the **unpooled sample proportions.**

This means the two sample proportions are treated separately.

(1.) The samples are simple random samples.

(2.) The population is normally distributed, OR the sample size is greater than thirty ($n \gt 30$).

Two samples are **independent** if the sample values from one population are not related to, or somehow naturally
paired or matched with the sample values from the other population.

__Example:__ Paul was curious about the mean credit scores of men and women.

He visited the *Town of Okay, Oklahoma* and gathered two random samples: the credit scores
of $50$ men and $50$ women.

Two samples are **dependent** if the sample values from one population are related to, or somehow naturally
paired or matched with the sample values from the other population.

Each pair of sample values consists of two measurements from the same subject (such as before/after data), or
each pair of sample values consists of matched pairs (such as husband/wife data).
__Example:__ Paul was curious about the mean credit scores of married couples.

He visited the *Community of Money, Mississippi,* randomly selected $50$ families, and asked for each of the credit
scores of the husband and wife.

When $\sigma_1$ and $\sigma_2$ are unknown, and are not assumed to be equal: AND

When $\mu_1$ and $\mu_2$ are assumed to be equal:

Use $t$ distribution

(1.) The values of the first population standard deviation and the second population standard deviation are unknown,
and are not assumed to be equal.

(2.) The two samples are simple random samples.

(3.) The two samples are independent.

(4.) The two samples are taken from a normally distributed population, or each of the samples have sizes greater than $30$.

When $\sigma_1$ and $\sigma_2$ are unknown, but assumed to be equal:

Use $t$ distribution and a pooled sample variance

(1.) The values of the first population standard deviation and the second population standard deviation are unknown,
but they are assumed to be equal.

(2.) The two samples are simple random samples.

(3.) The two samples are independent.

(4.) The two samples are taken from a normally distributed population, or each of the samples have sizes greater than $30$.

When $\sigma_1$ and $\sigma_2$ are known:

Use $z$ distribution

(1.) The values of the first population standard deviation and the second population standard deviation are known.

(2.) The two samples are simple random samples.

(3.) The two samples are independent.

(4.) The two samples are taken from a normally distributed population, or each of the samples have sizes greater than $30$.

Use $t$ distribution for Dependent Samples

(1.) The samples are dependent samples (matched pairs).

(2.) The two samples are simple random samples.

(3.) The two samples are taken from a normally distributed population, or each of the samples have sizes greater than $30$.

Use $\chi^2$ distribution

(1.) The samples are simple random samples.

(2.) The population is normally distributed.

This is used to test the hypothesis that an observed frequency distribution **fits** to some claimed distribution.

It is used for analyzing categorical or quanlitative data that can be separated into different rows and columns of a table.

$H_0:$ The frequency counts agree with the claimed distribution

$H_1:$ The frequency counts do not agree with the claimed distribution.

A **hypothesis** is **a claim about a population parameter.**

A **hypothesis test** is a **procedure for testing a claim about a population parameter.**

The **null hypothesis** is the **statement** that shows that the value of the **population parameter is equal to some claimed value.**

The **alternative hypothesis** is the **statement** that shows that the value of the **population parameter is different from the claimed value.**

A **hypothesis test** is a **left-tailed test** if the **population parameter is less than the claimed value.**

A **hypothesis test** is a **right-tailed test** if the **population parameter is greater than the claimed value.**

A **hypothesis test** is a **two-tailed test** if the **population parameter is not equal than he claimed value.**

The **test statistic** is a **used in making a decision about the null hypothesis.**

**Type I Error** is the error made when we reject the null hypothesis when it is true.

**Type II Error** is the error made when we fail to reject the null hypothesis when it is false.

The **level of significance** is the **probability of making the mistake of rejecting the null hypothesis when it is true.**

This implies that: The **level of significance** is the **probability of making a Type I error.**

The **power of a hypothesis test** is the **probability of rejecting a null hypothesis when it is false.**

This implies that: The **power of a hypothesis test** is the **probability of making the correct decision by avoiding making a Type II error.**

Two samples are **independent** if the sample values from one population are not related to, or somehow naturally
paired or matched with the sample values from the other population.

Two samples are **dependent** if the sample values from one population are related to, or somehow naturally
paired or matched with the sample values from the other population.

- $z_{\dfrac{\alpha}{2}}$ is the critical $z$ value
- $z_{\dfrac{\alpha}{2}}$ is the $z-score$ separating an area/probability of $\dfrac{\alpha}{2}$ in the right tail
- $-z_{\dfrac{\alpha}{2}}$ is the $z-score$ separating an area/probability of $\dfrac{\alpha}{2}$ in the left tail
- $z_{\alpha}$ is the $z-score$ separating an area/probability of $\alpha$ in the right tail
- $-z_{\alpha}$ is the $z-score$ separating an area/probability of $\alpha$ in the left tail
- $t_{\dfrac{\alpha}{2}}$ is the critical $t$ value
- $t_{\dfrac{\alpha}{2}}$ is the critical $t$ separating an area/probability of $\dfrac{\alpha}{2}$ in the right tail
- $-t_{\dfrac{\alpha}{2}}$ is the critical $t$ separating an area/probability of $\dfrac{\alpha}{2}$ in the left tail
- $z$ is the test statistic for estimating population proportion
- $z$ is the test statistic for estimating population mean (based on some conditions)
- $t$ is the test statistic for estimating population mean (based on some conditions)
- $\chi ^2$ is the test statistic for estimating population variance
- $\chi ^2$ is the test statistic for estimating population standard deviation
- $\chi ^2$ is the test statistic for Goodness-of-Fit tests
- $\alpha$ is the level of significance
- $\alpha$ is the probability of making a Type I error
- $\beta$ is the probability of making a Type II error
- $CL$ is the level of confidence
- $df$ is the degrees of freedom
- $P-value$ is the probability value
- $\hat{p}$ is the sample proportion
- $x$ is the number of individuals with the specified characteristics
- $n$ is the sample size
- $n$ is the total number of trials
- $p$ is the population proportion
- $q$ is the complement of the population proportion
- $\overline{p}$ is the pooled sample proportion
- $\overline{q}$ is the complement of the pooled sample proportion
- $s$ is the sample standard deviation
- $s^2$ is the sample variance
- $\sigma$ is the population standard deviation
- $\sigma^2$ is the population variance
- $x_1$ is the number of successes in the first sample
- $x_2$ is the number of successes in the second sample
- $\hat{p_1}$ is the first sample proportion
- $\hat{p_2}$ is the second sample proportion
- $\hat{q_1}$ is the complement of the first sample proportion
- $\hat{q_2}$ is the complement of the second sample proportion
- $\hat{p_c}$ is the critical value of the sample proportion
- $p_a$ is the alternative proportion
- $q_a$ is the complement of the alternative proportion
- $p_1$ is the first population proportion
- $p_2$ is the second population proportion
- $SE$ is the standard error
- $E$ is the margin of error
- $\overline{x_1}$ is the first sample mean
- $\overline{x_2}$ is the second sample mean
- $\overline{x_c}$ is the critical value of the sample mean
- $\overline{x_a}$ is the alternative mean
- $\mu_1$ is the first population mean
- $\mu_2$ is the second population mean
- $s_1$ is the first sample standard deviation
- $s_2$ is the second sample standard deviation
- $\sigma_1$ is the first population standard deviation
- $\sigma_2$ is the second population standard deviation
- $s_1^2$ is the first sample variance
- $s_2^2$ is the second sample variance
- $s_p^2$ is the pooled sample variance
- $\sigma_1^2$ is the first population variance
- $\sigma_2^2$ is the second population variancen
- $d$ is the individual difference between the two values in a single matched pair
- $\mu_d$ is the mean of the differences for the population of all matched pairs of data
- $\overline{d}$ is the mean value of the differences for the paired sample data
- $s_d$ is the standard deviation of the differences for the paired sample data
- $n_d$ is number of pairs of sample data
- $Ob$ is the observed frequence of an outcome (found from the sample data)
- $Ex$ is the expected frequency of an outcome (found by assuming the distribution as claimed)
- $k$ is the number of different categories
- $r$ is the sample Pearson's linear correlation coefficient
- $\rho$ is the population Pearson's linear correlation coefficient

Proportion is given as a decimal or a percentage.

If the significance level, $\alpha$ is not given, use $\alpha = 5\%$

The null hypothesis should always have the 'equal' symbol

The alternative hypothesis has the 'unequal' symbol (less than, greater than, not equal to)

To calculate the probability of making a Type II error, $\beta$

**As applicable:**

**First Step:**

Solve for the critical proportion using the critical $z$ OR

Solve for the critical mean using the critical $z$ OR

Solve for the critical mean using the critical $t$

$
(1.)\:\: z_{\dfrac{\alpha}{2}} = \dfrac{\hat{p_c} - p}{\sqrt{\dfrac{pq}{n}}} \\[10ex]
(2.)\:\: \hat{p_c} = z_{\dfrac{\alpha}{2}} * \sqrt{\dfrac{pq}{n}} + p \\[7ex]
(3.)\:\: z_{\dfrac{\alpha}{2}} = \dfrac{\overline{x}_c - \mu}{\dfrac{\sigma}{\sqrt{n}}} \\[10ex]
(4.)\:\: \overline{x_c} = z_{\dfrac{\alpha}{2}} * \dfrac{\sigma}{\sqrt{n}} + \mu \\[7ex]
(5.)\:\: t_{\dfrac{\alpha}{2}} = \dfrac{\overline{x}_c - \mu}{\dfrac{s}{\sqrt{n}}} \\[10ex]
(6.)\:\: \overline{x_c} = t_{\dfrac{\alpha}{2}} * \dfrac{s}{\sqrt{n}} + \mu \\[7ex]
$
**Second Step:**

Solve for the $z$ score using the alternative proportion OR

Solve for the $z$ score using the alternative mean OR

Solve for the $t$ score using the alternative mean

$
(7.)\:\: z = \dfrac{\hat{p_c} - p_a}{\sqrt{\dfrac{p_a * q_a}{n}}} \\[10ex]
(8.)\:\: z = \dfrac{\overline{x}_a - \mu}{\dfrac{\sigma}{\sqrt{n}}} \\[10ex]
(9.)\:\: t = \dfrac{\overline{x}_a - \mu}{\dfrac{s}{\sqrt{n}}} \\[10ex]
$
**Third Step:**

Calculate the probability of $z$ OR

Calculate the probability of $t$

$ (1.)\:\: P_{hyp} = 1 - \beta \\[3ex] $

If $p$ is not given, use $p = 50\%$

$ (1.)\:\: \hat{p} = \dfrac{x}{n} \\[7ex] (2.)\:\: p + q = 1 \\[5ex] (3.)\:\: z = \dfrac{\hat{p} - p}{\sqrt{\dfrac{pq}{n}}} \\[10ex] $

The pooled sample proportion is used.

This means that the data is pooled to obtain the proportion of successess in both samples combined
rather than separately (unpooled).

$ (1.)\:\: \hat{p_1} = \dfrac{x_1}{n_1} \\[7ex] (2.)\:\: \hat{p_2} = \dfrac{x_2}{n_2} \\[7ex] (3.)\:\: \overline{p} = \dfrac{x_1 + x_2}{n_1 + n_2} \\[7ex] (4.)\:\: \overline{q} = 1 - \overline{p} \\[5ex] (5.)\:\: z = \dfrac{(\hat{p_1} - \hat{p_2}) - (p_1 - p_2)}{\sqrt{\dfrac{\overline{p} * \overline{q}}{n_1} + \dfrac{\overline{p} * \overline{q}}{n_2}}} \\[10ex] $

Because this is a pooled sample, we assume the null hypothesis as:

$
H_0:\:\: p_1 = p_2 \:\:OR \\[3ex]
H_0:\:\: p_1 - p_2 = 0 \\[3ex]
\implies \\[3ex]
(6.)\:\: z = \dfrac{\hat{p_1} - \hat{p_2}}{\sqrt{\dfrac{\overline{p} * \overline{q}}{n_1} +
\dfrac{\overline{p} * \overline{q}}{n_2}}} \\[10ex]
(7.)\:\: SE = \sqrt{\dfrac{\overline{p} * \overline{q}}{n_1} + \dfrac{\overline{p} *
\overline{q}}{n_2}} \\[7ex]
$

The unpooled sample proportion is used.

This means that the two sample proportions are treated separately.

$
(1.)\:\: E = z_{\dfrac{\alpha}{2}} * \sqrt{\dfrac{\hat{p_1} * \hat{q_1}}{n_1} + \dfrac{\hat{p_2} *
\hat{q_2}}{n_2}} \\[7ex]
$
The confidence interval method of the difference of the population proportions is:

$(\hat{p_1} - \hat{p_2}) - E \lt (p_1 - p_2) \lt (\hat{p_1} - \hat{p_2}) + E$

**When $\sigma$ is known:**

$
(1.)\:\: z = \dfrac{\overline{x} - \mu}{\dfrac{\sigma}{\sqrt{n}}} \\[10ex]
$
**When $\sigma$ is NOT known:**

$
(2.)\:\: z = \dfrac{\overline{x} - \mu}{\dfrac{s}{\sqrt{n}}} \\[10ex]
(3.)\:\: df = n - 1 \\[3ex]
$

Two samples are **independent** if the sample values from one population are not related to, or
natuarlly paired or
paired with the sample values from the other population.

**
When $t$ distribution is used
When $\sigma_1$ and $\sigma_2$ are unknown; and are NOT assumed to be equal
**

$ (1.)\:\: t = \dfrac{(\overline{x_1} - \overline{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}} \\[10ex] $

$ (2.)\:\: t = \dfrac{\overline{x_1} - \overline{x_2}}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}} \\[10ex] $

$ (3.)\:\: df = the\:\:smaller\:\:value\:\:of\:\:(n_1 - 1) \:\:and\:\: (n_2 - 1) \\[3ex] $

$ (4.)\:\: df = \dfrac{(A + B)^2}{\dfrac{A^2}{n_1 - 1} + \dfrac{B^2}{n_2 - 1}} \\[10ex] (5.)\:\: A = \dfrac{s_1^2}{n_1} \\[7ex] (6.)\:\: B = \dfrac{s_2^2}{n_2} \\[7ex] $

When $\sigma_1$ and $\sigma_2$ are unknown; and are assumed to be equal

The pooled dample variance is used.

$ (1.)\:\: t = \dfrac{(\overline{x_1} - \overline{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{s_p^2}{n_1} + \dfrac{s_p^2}{n_2}}} \\[10ex] (2.)\:\: s_p^2 = \dfrac{s_1^2(n_1 - 1) + s_2^2(n_2 - 1)}{(n_1 - 1)(n_2 - 1)} \\[7ex] (3.)\:\: s_p = \sqrt{\dfrac{s_1^2(n_1 - 1) + s_2^2(n_2 - 1)}{(n_1 - 1)(n_2 - 1)}} \\[7ex] (4.)\:\: df = n_1 + n_2 - 2 \\[3ex] $

When $\sigma_1$ and $\sigma_2$ are known

$ (1.)\:\: z = \dfrac{(\overline{x_1} - \overline{x_2}) - (\mu_1 - \mu_2)}{\sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}}} \\[10ex] $

**When $\sigma_1$ and $\sigma_2$ are unknown; and are NOT assumed to be equal**

The confidence interval method of the difference of the population means is:

$
(1.)\:\: Confidence\:\:Interval\:\:is:\:\:(\overline{x_1} - \overline{x_2}) - E \lt (\mu_1 - \mu_2)
\lt (\overline{x_1} - \overline{x_2}) + E \\[5ex]
(2.)\:\: E = t_{\dfrac{\alpha}{2}} * \sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}} \\[7ex]
$
**Simple Estimate of the Degrees of Freedom:**

$
(3.)\:\: df = the\:\:smaller\:\:value\:\:of\:\:(n_1 - 1) \:\:and\:\: (n_2 - 1) \\[3ex]
$
**Difficult Estimate (More Accurate Estimate) of the Degrees of Freedom:**

$
(4.)\:\: df = \dfrac{(A + B)^2}{\dfrac{A^2}{n_1 - 1} + \dfrac{B^2}{n_2 - 1}} \\[10ex]
(5.)\:\: A = \dfrac{s_1^2}{n_1} \\[7ex]
(6.)\:\: B = \dfrac{s_2^2}{n_2} \\[7ex]
$
**When $\sigma_1$ and $\sigma_2$ are unknown; and are NOT assumed to be equal**

The confidence interval method of the difference of the population means is:

$
(1.)\:\: Confidence\:\:Interval\:\:is:\:\:(\overline{x_1} - \overline{x_2}) - E \lt (\mu_1 - \mu_2)
\lt (\overline{x_1} - \overline{x_2}) + E \\[5ex]
(2.)\:\: E = t_{\dfrac{\alpha}{2}} * \sqrt{\dfrac{s_p^2}{n_1} + \dfrac{s_p^2}{n_2}} \\[7ex]
(3.)\:\: s_p^2 = \dfrac{s_1^2(n_1 - 1) + s_2^2(n_2 - 1)}{(n_1 - 1)(n_2 - 1)} \\[7ex]
(4.)\:\: s_p = \sqrt{\dfrac{s_1^2(n_1 - 1) + s_2^2(n_2 - 1)}{(n_1 - 1)(n_2 - 1)}} \\[7ex]
(5.)\:\: df = n_1 + n_2 - 2 \\[3ex]
$
**
When $z$ distribution is used
When $\sigma_1$ and $\sigma_2$ are known
**

The confidence interval method of the difference of the population means is:

$ (1.)\:\: Confidence\:\:Interval\:\:is:\:\:(\overline{x_1} - \overline{x_2}) - E \lt (\mu_1 - \mu_2) \lt (\overline{x_1} - \overline{x_2}) + E \\[5ex] (2.)\:\: E = z_{\dfrac{\alpha}{2}} * \sqrt{\dfrac{\sigma_1^2}{n_1} + \dfrac{\sigma_2^2}{n_2}} \\[7ex] $

Two samples are **dependent** if the sample values are somehow matched, where the matching is
based on some meaningful
relationship.

Each pair of sample values consists of two measurements from the same subject (such as
**before/after** data), or each pair
of sample values consists of matched pairs (such as **husband/wife** data).

$
(1.)\:\: t = \dfrac{\overline{d} - \mu_d}{\dfrac{s_d}{\sqrt{n}}} \\[7ex]
(2.)\:\: df = n - 1
$

$ (1.)\:\: Confidence\:\:Interval\:\:is:\:\: \overline{d} - E \lt \mu_d \lt \overline{d} + E \\[5ex] (2.)\:\: E = t_{\dfrac{\alpha}{2}} * \dfrac{s_d}{\sqrt{n}} \\[5ex] $

$ (1.)\:\: \chi^2 = \Sigma \dfrac{(Ob - Ex)^2}{Ex} \\[7ex] (2.)\:\: df = k - 1 \\[3ex] $

**Two Samples**

**First Formula for the Pearson Correlation Coefficient**

$
(1.)\:\: r = \dfrac{\Sigma \left(\dfrac{x - \overline{x}}{s_x}\right)\left(\dfrac{y -
\overline{y}}{s_y}\right)}{n - 1} \\[10ex]
(2.)\:\: r = \dfrac{\Sigma(z_x)(z_y)}{n - 1} \\[5ex]
$
**Second Formula for the Pearson Correlation Coefficient**

$
(1.)\:\: r = \dfrac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{n(\Sigma x^2) - (\Sigma x)^2} *
\sqrt{n(\Sigma y^2) - (\Sigma y)^2}} \\[7ex]
$
**Critical Value of the Correlation Coefficient** (Use for Critical Value method)

$
(1.)\:\: Critical\:\:value\:\:of\:\:r = \sqrt{\dfrac{t^2}{t^2 + df}} \\[7ex]
(where\:\:t-values\:\:are\:\:from\:\:the\:\:Critical\:\:t\:\:Table) \\[3ex]
(2.)\:\: df = n - 2 \\[3ex]
$
**Test Statistic of the Correlation Coefficient** (Use for P-value method)

$
(1.)\:\: t = \dfrac{r}{\sqrt{\dfrac{1 - r^2}{n - 2}}} \\[7ex]
$

**NOTE:**

(1.) Use $z$ or $t$ as necessary

(2.) For the interpretations, replace **proportion** with **mean** as applicable.

__Unless your professor says otherwise:__

(3.) Use only these terms for the **Decision**

(a.) Reject the null hypothesis

(b.) Do not reject the null hypothesis.

(c.) Fail to reject the null hypothesis.

(b.) and (c.) means the same thing.

*
Keep in mind that we are dealing with the null hypothesis
You either reject the null hypothesis or you do not reject the null hypothesis.
We do not accept the null hypothesis.
We just do not reject it.
The fact that we do not reject it does not mean that we accept it.
Hence, the need for the conclusion and the interpretation.
*

(4.) Use only these terms for the

(a.) There is sufficient evidence to support/warrant the rejection of the null hypothesis.

(b.) There is sufficient evidence to reject the null hypothesis.

(c.) There is sufficient evidence to support/warrant the claim of the alternative hypothesis.

(d.) There is insufficient evidence to support/warrant the rejection of the null hypothesis.

(e.) There is insufficient evidence to reject the null hypothesis.

(f.) There is insufficient evidence to support/warrant the claim of the alternative hypothesis.

(a.), (b.), and (c.) means the same thing.

(d.), (e.), and (f.) means the same thing.

**
IMPORTANT:
(1.) For:
Hypothesis Test about a Population Proportion OR
Hypothesis Test about a Population Mean
Use $z$ table
(2.) For:
Hypothesis Test about a Population Mean (whose Population Standard Deviation is not known)
Use $t$ table
(3.) For:
Hypothesis Test about a Population Variance OR
Hypothesis Test about a Population Standard Deviation
use $\chi^2$ table
**

Find $-z_{\dfrac{\alpha}{2}}$

**Condition $1$**

If $z \lt -z_{\dfrac{\alpha}{2}}$, it falls in the critical region

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **significantly less** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **significantly less** than the
population proportion for the *second variable*

**Condition $2$**

If $z \gt -z_{\dfrac{\alpha}{2}}$, it does not fall in the critical region

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **NOT significantly less** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **NOT significantly less** than the
population proportion for the *second variable*

Find $z_{\dfrac{\alpha}{2}}$

**Condition $1$**

If $z \gt z_{\dfrac{\alpha}{2}}$, it falls in the critical region

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **significantly more** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **significantly more** than the
population proportion for the *second variable*

**Condition $2$**

If $z \lt z_{\dfrac{\alpha}{2}}$, it does not fall in the critical region

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **NOT significantly more** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **NOT significantly more** than the
population proportion for the *second variable*

Find $-z_{\dfrac{\alpha}{2}}$ and $z_{\dfrac{\alpha}{2}}$

**Condition $1$**

If $z \lt -z_{\dfrac{\alpha}{2}}$ OR $z \gt z_{\dfrac{\alpha}{2}}$, it falls in the critical region

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to support the *rejection of the null hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **significantly different** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **significantly different** from the
population proportion for the *second variable*

**Condition $2$**

If $-z_{\dfrac{\alpha}{2}} \lt z \lt z_{\dfrac{\alpha}{2}}$, it does not fall in the critical region

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to warrant the *rejection of the null hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **NOT significantly different** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **NOT significantly different** from the
population proportion for the *second variable*

Find $P(z \lt -test\:\:statistic)$

**Condition $1$**

If $P-value \le \alpha$

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **significantly less** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **significantly less** than the
population proportion for the *second variable*

**Condition $2$**

If $P-value \gt \alpha$

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **NOT significantly less** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **NOT significantly less** than the
population proportion for the *second variable*

Find $P(z \gt test\:\:statistic)$

**Condition $1$**

If $P-value \le \alpha$

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **significantly more** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **significantly more** than the
population proportion for the *second variable*

**Condition $2$**

If $P-value \gt \alpha$

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to support the *claim of the alternative hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **NOT significantly more** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **NOT significantly more** than the
population proportion for the *second variable*

Find $P(z \lt -test\:\:statistic) + P(z \gt test\:\:statistic)$

**Condition $1$**

If $P-value \le \alpha$

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to support the *rejection of the null hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **significantly different** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **significantly different** from the
population proportion for the *second variable*

**Condition $2$**

If $P-value \gt \alpha$

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to warrant the *rejection of the null hypothesis*

**Interpretation for One-Sample:** The population proportion for the *variable* is **NOT significantly different** than the stated value.

**Interpretation for Two-Samples:** The population proportion for the *first variable* is **NOT significantly different** from the
population proportion for the *second variable*

Construct a confidence interval using: $CL = 1 - 2\alpha$

Determine the confidence interval using an appropriate confidence level

**Condition $1$**

If the confidence interval **does NOT** the value of the population parameter stated in the null hypothesis:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *warrant the rejection of the null hypothesis*

**Interpretation:** The confidence interval **does NOT** contain the value of the *population parameter* stated in
the null hypothesis

**Condition $2$**

If the confidence interval contains the value of the population parameter stated in the null hypothesis:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of the alternative hypothesis*

**Interpretation:** The confidence interval contains the value of the *population parameter* stated in
the null hypothesis

Construct a confidence interval using: $CL = 1 - \alpha$

Determine the confidence interval using an appropriate confidence level

**Condition $1$**

If $Lower\:\:Confidence\:\:Limit \lt 0$ AND $Upper\:\:Confidence\:\:Limit \lt 0$:
the confidence interval does not contain $0$

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of the alternative hypothesis*

**Interpretation:** The population proportion for the *first variable* is **significantly less** than the
population proportion for the *second variable*

**Condition $2$**

If $Lower\:\:Confidence\:\:Limit \lt 0$ AND $Upper\:\:Confidence\:\:Limit \gt 0$:
the confidence interval contains $0$

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of the alternative hypothesis*

**Interpretation:** The population proportion for the *first variable* is **NOT significantly less** than the
population proportion for the *second variable*

Construct a confidence interval using: $CL = 1 - \alpha$

Determine the confidence interval using an appropriate confidence level

**Condition $1$**

If $Lower\:\:Confidence\:\:Limit \gt 0$ AND $Upper\:\:Confidence\:\:Limit \gt 0$:
the confidence interval does not contain $0$

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of the alternative hypothesis*

**Interpretation:** The population proportion for the *first variable* is **significantly more** than the
population proportion for the *second variable*

**Condition $2$**

If $Lower\:\:Confidence\:\:Limit \lt 0$ AND $Upper\:\:Confidence\:\:Limit \gt 0$:
the confidence interval contains $0$

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of the alternative hypothesis*

**Interpretation:** The population proportion for the *first variable* is **NOT significantly more** than the
population proportion for the *second variable*

Construct a confidence interval using: $CL = 1 - \alpha$

Determine the confidence interval using an appropriate confidence level

**Condition $1$**

If $Lower\:\:Confidence\:\:Limit \lt 0$ AND $Upper\:\:Confidence\:\:Limit \lt 0$ OR
$Lower\:\:Confidence\:\:Limit \gt 0$ AND $Upper\:\:Confidence\:\:Limit \gt 0$
the confidence interval does not contain $0$

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *warrant the rejection of the null hypothesis*

**Interpretation:** The population proportion for the *first variable* is **significantly different** from the
population proportion for the *second variable*

**Condition $2$**

If $Lower\:\:Confidence\:\:Limit \lt 0$ AND $Upper\:\:Confidence\:\:Limit \gt 0$:
the confidence interval contains $0$

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *warrant the rejection of the null hypothesis*

**Interpretation:** The population proportion for the *first variable* is **NOT significantly different** from the
population proportion for the *second variable*

Calculate the Pearson correlation coefficient

Calculate the critical value of the Pearson correlation coefficient

**Condition $1$**

If $r \lt 0$ AND $r \lt Critical\:\:value\:\:of\:\:r$:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of a negative linear correlation*

**Condition $2$**

If $r \lt 0$ AND $r \gt Critical\:\:value\:\:of\:\:r$:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of a negative linear correlation*

Calculate the Pearson correlation coefficient

Calculate the critical value of the Pearson correlation coefficient

**Condition $1$**

If $r \gt 0$ AND $r \gt Critical\:\:value\:\:of\:\:r$:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of a positive linear correlation*

**Condition $2$**

If $r \gt 0$ AND $r \lt Critical\:\:value\:\:of\:\:r$:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of a positive linear correlation*

Calculate the Pearson correlation coefficient

Calculate the critical value of the Pearson correlation coefficient

**Condition $1$**

If $|r| \gt Critical\:\:value\:\:of\:\:r$:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of a linear correlation*

**Condition $2$**

If $|r| \le Critical\:\:value\:\:of\:\:r$:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of a linear correlation*

Calculate the Pearson correlation coefficient

Calculate the test statistic

Determine the probability value (p-value) of the test statistic

**Condition $1$**

If $r \lt 0$ AND $P-value \le \alpha$:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of a negative linear correlation*

**Condition $2$**

If $r \lt 0$ AND $P-value \gt \alpha$:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of a negative linear correlation*

Calculate the Pearson correlation coefficient

Calculate the test statistic

Determine the probability value (p-value) of the test statistic

**Condition $1$**

If $r \gt 0$ AND $P-value \le \alpha$:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of a positive linear correlation*

**Condition $2$**

If $r \gt 0$ AND $P-value \gt \alpha$:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of a positive linear correlation*

Calculate the Pearson correlation coefficient

Calculate the test statistic

Determine the probability value (p-value) of the test statistic

**Condition $1$**

If $P-value \le \alpha$:

**Decision:** Reject the null hypothesis

**Conclusion:** There is sufficient evidence to *support the claim of a linear correlation*

**Condition $2$**

If $P-value \gt \alpha$:

**Decision:** Do not reject the null hypothesis

**Conclusion:** There is insufficient evidence to *support the claim of a linear correlation*

Chukwuemeka, S.D (2017, April 30). *Samuel Chukwuemeka Tutorials - Math, Science, and Technology.*
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Kozak, K. (2014). Statistics Using Technology.

Sullivan, M., Barnett, R. A., Ziegler, M. R., & Byleen, K. E. (2013). Statistics : informed decisions using data with an introduction to mathematics of finance. Pearson Learning Solutions.

Triola, M. F., & Iossi, L. (2019). Elementary statistics using the TI-83/84 plus calculator. Pearson.

Sullivan, M., & Sullivan, M. (2017). *Algebra & Trigonometry* ($7^{th}$ ed.).
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Radford University – Table of Critical Values for Pearson's r. (n.d.). Retrieved April 30, 2018, from https://www.radford.edu/~jaspelme/statsbook/Chapter%20files/Table_of_Critical_Values_for_r.pdf