Introduction

BA501 : The Class : Stats : Hypothesis : Introduction
Hypothesis Testing: Introduction
Hypothesis testing is the second method of statistical inference

Introduction- Lesson 1

§ Introduction §

Hypothesis testing is the second method of statistical inference. Confidence interval calculation was the first. When using confidence intervals, there is an attempt to quantify the value of the unknown population parameter (m in our case) over a range of possible values based on a point estimate (xbar). Hypothesis testing is designed to estimate the probability of the sample result (xbar) occurring by chance, if the hypothesized value of the population parameter (the null hypothesis, H_o) is in fact correct. If the statistical distance between xbar and m is large, we will reject the H_o. The alternative hypothesis, (H_a) which is the opposite of the H_o is then supported. If the statistical distance between xbar and m is small, we will support the H_o.

§ Hypothesis § Null Hypothesis § Alternative Hypothesis §

§ Mistakes § Correct Decisions § Decision Table §

§ Five-Step Procedure § Confidence Intervals §

§ One-Tailed Test § p-Values §

1. Hypothesis-

An hypothesis is a claim, speculation, well educated guess, or testable hypothesis. When a testable hypothesis is evaluated with data, one can not prove that it is correct (or true). The researcher can find evidence to reject the hypothesis or to support (not prove) the hypothesis. One can "rule out" (reject) a hypothesis, but one can not prove that it is correct. (click me)

An hypothesis test is like a blood test to establish paternity, it can rule out but it can not rule in. True False (click one)

An hypothesis must be _________ or it is true by definition..

2. Null Hypothesis ( H_o )-

A statement concerning a population parameter, or other non-mathematical statement. The researcher wishes to reject H_o. (click me)

The null hypothesis is assumed to be true unless shown to be otherwise. True False (click one)

A null hypothesis assumes that nothing ________ is going on i.e. that its hypothesized value is correct.

3. Alternative Hypothesis ( H_a )-

The opposite of the null hypothesis. The hypothesis is supported if the null hypothesis is rejected. The researcher wishes to support H_a(also called the research hypothesis). (click me)

The alternative hypothesis has the same mathematical sign as null hypothesis. True False (click one)

The alternative hypothesis is ____ assumed to be correct.

4. Mistakes: Type I and Type II Errors- since the total population is not used, there is always a chance that we have drawn the wrong conclusion based on the sample.

(A) a = the probability of rejecting H_o when H_o is true = Type I Error (click me)

[1] This is the same a used in earlier material, a = the probability of not capturing m in a CI. In this chapter, a is called the "level of significance."

(B) b = the probability of Failing To Reject ( support ) H_o when H_o is false = Type II Error (click me)

The Type I error = a can be reduced to zero. True False (click one)

If Ho is false but not known the the researcher, then one is likely to comment a Type ___ error.

5. Correct Decisions

(A) Reject H_o when H_o is false; correct decision

(B) FTR ( support ) H_o when H_o is true; correct decision (FTR = Fail To Reject) (click me)

The researcher wishes to do which of the above? A B (click one)

One is never really ____ that the correct statistical decision has been made.

6. Decision Table

Statistical

Decision

Actual Situation

Reject H_o

FTR(support) H_o

H_o True

Type I Error (a)

correct

H_o False

correct

Type II Error (b)

The actual situation is always know. True False (click one)

It is ____ possible to comment Type I and Type II errors at the same time.

7. The Five-Step Procedure for Hypothesis Testing- For a two-tail test

(A) Set up the Null Hypothesis, H_o, and Alternative Hypothesis, H_a.

H_o: µ = µ_o

H_a: µ ¹ µ_o (click me)

[1] The number to the right of the equal sign is always the claimed parameter value (never xbar ). This is true for both H_o and H_a. (click me)

[2] The signs used in H_o and H_a are always the mathematical opposites of each other. Some form of an "equal sign" ( = ) is always associated with the H_o (never H_a). (click me)

[3] Question: Is our statistic, xbar "close enough" to the claimed parameter value, µ_o, to FTR (support) H_o?

[a] We do not expect our estimate, xbar, to be exactly equal to the claimed parameter value, µ_o. (click me)

[b] Is xbar "statistically close enough" to m_o to believe the H_o? Use the standard error, s_xbar or s _xbar, and Z or t score to measure the statistical distance between them.

The sample mean, xbar, is placed in the middle of the distribution to start a hypothesis test. True False (click one)

An equal sign is never associated with the _____ hypothesis.

(B) Define the test statistic. Use Z or t?

[1] Source of Standard Deviation- population or sample?

[2] Degrees of Freedom (n - 1)- Large or Small? (click me)

		Population s	Sample s
Sample Size	Small df £ 30	Z	t
Sample Size	Large df > 30	Z	Z or t

[3] Use z or t to calculate the size of the difference between xbar and µ_o.

The decision matrix presented here is the same as that used with confidence intervals. True False (click one)

Then the degrees of freedom are _____ it allows one to use either Z or t.

(C) Define a region(s) of rejection based on a.

[1] What a can be tolerated? What percent of the time will you reject H_o when H_o is true?

[2] Use a = 0.05 when in doubt. (click me)

[3] For a two - tail test, each tail will have a / 2 in it.

[a] a/2 = 0.025

[b] 0.5 - a/2 = 0.5 - 0.025 = 0.4750 (look up in Z table)

[4] table value of test statistic ± Z_0.025 = ± 1.96 ( critical value) (click me)

[a] The table value is always associated with a (never with xbar), is found in a Z (or t ) table with subscript a or a/2, and defines the region(s) of rejection.

The alpha used with confidence intervals and the alpha use in hypothesis test are different alphas. True False (click one)

For a two-tail hypothesis test, what Z score associated with a = 0.10.

(D) Calculate the value of the test statistic and carry out the test

[1] The calculated (or computed ) value of the test statistic is always associated with xbar (never a), has a "star" next to it ( Z* or t*), and is calculated by you (not found in a table).

[2] Z* = [xbar - m_o ] / [s /Ö n] or

t* = [xbar - m_o ] / [s /Ö n] (click me)

[3] and carry out the test:

Large-Sample Two-Tailed Tests on the Population Mean

H_o: µ = µ_o

H_a: µ ¹ µ_o

Reject H_o if |Z*| > Z a _/2

FTR (support) H_o if |Z*| £ Z a _/2

Taking the absolue value of the computed value of the test statistic (Z* or t*) forces two-tail tests into the left end of the distribution. True False (click one)

When the computed value of the test statistic is equal to the table (critical) value, one must _____ the null hypothesis, Ho.

(E) Give a conclusion in terms of the original problem or question. (click me)

8. Confidence Intervals and Hypothesis Testing

(A) A confidence interval can be used to do hypothesis testing for two -tailed tests (only). (click me)

(B) a is split into two parts (a / 2) for both CIs and two-tailed hypothesis test.

(C) Rules:

(1) If the claimed µ_o lies within the CI, then (FTR) support H_o.

(2) If the claimed µ_o lies outside the CI, then reject H_o

The use of confidence intervals to do hypothesis tests applies to one-tail test in addition to two-tail test. True False (click one)

The results of an hypothesis test using a confidence interval may change if _____ (probability of not capturing hypothesized parameter value) changes.

9. One-Tailed Test for the Mean of a Population (both Z and t)

(A) The major difference of two-tailed and one-tailed hypothesis tests is:

(1) two-tailed, split to give a / 2 in each tail.

(2) one-tailed, do not split. Put entire a into left tail or right tail.

(a) Put in the left tail when H_a has an "less than" sign (<) (click me)

(b) Put in the right tail when H_a has an "greater than" sign (>) (click me)

(B) Rules:

Large-Sample One-Tailed (left) Tests on the Population Mean

H_o: µ ³ µ_o

H_a: µ < µ_o

Reject H_o if Z* < - Za

FTR(Support) H_o if Z* ³ - Za

Large-Sample One-Tailed (right) Tests on the Population Mean

H_o: µ £ µ_o

H_a: µ > µ_o

Reject H_o if Z* > Za

FTR(Support) H_o if Z* £ Za

On which end of the distribution would xbar fall in order to find extreme statistical evidence against a H_o: µ ³ µ_o? Right Left (click one)

In the question above, all of alpha to form the region of rejection this place in the ______ tail of the distribution.

10. Reporting Testing Results Using a p-Values

(A) The p-value is the probability associated with the calculated ( computed ) Z* (or t*).

(1) It is the area in the tail of the distribution beyond Z* or t*.

(2) Procedure for Finding the p-value:

(a) For H_a: m ¹ m _o, p-value = ( 2 )(area outside Z* or t*) (click me)

(b) For H_a: m > m _o, p-value = area to right of Z* or t* (click me)

(c) For H_a: m < m _o, p-value = area to left of Z* or t* (click me)

(3) The p-value is the value of a at which the hypothesis test procedure changes conclusions based on a given set of data. (click me)

(4) It is the largest value of a for which you will FTR ( support ) H_o. (click me)

A p-value is a probability found under the curve of a distribution just like any other probability. True False (click one)

Suppose a p-value calculation gives the probability to the left end of the distribution. This indicates an alternative hypothesis test with a _____ _____ inequality.

(B) Rules when a is given:

(1) Reject H_o if p-value < a

(a) If the p-value < a , then Z* or t* does falls in the tail created by the table Z or t; therefore, Reject H_o.

(click me)

(2) FTR ( support ) H_o if p-value ³ a

(a) If the p-value ³ a , then Z* or t* does not fall in the tail created by the table Z or t; therefore, FTR (support) H_o.

(click me)

P-values and alphas are both probabilities. True False (click one)

When alpha equals the p-value, one must _____ the null hypothesis, Ho.

(C) If a is not known, use the General Rules of Thumb:

(1) Reject H_o if p-value is small (p-value < 0.01) (click me)

(a) Most levels of significance (a) chosen are > 0.01; therefore, when p-value < 0.01, reject H_o, since p-value would be < a.

(2) FTR ( support ) H_o if p-value is large (p-value > 0.10) (click me)

(a) Most levels of significance (a) chosen are £ 0.10; therefore, when p-value > 0.10, FTR( support ) H_o, since p-value would be > a.

(3) Test inconclusive if: (0.01 £ p-value £ 0.10 )

(a) Most levels of significance (a) chosen are between 0.01 and 0.10; therefore, the test is inconclusive, since the p-value could be on either side of a. (click me)

A small p-value indicates that the computed value of the test statistic (Z* or t*) would most likely not fall in the tail created by the table (critical) value of the test statistic (if provided). True False (click one)

When a p-value is ______ it would be most likely that computed value of the test statistic (Z* or t*) would not fall in the tail created by the table (critical) value of the test statistic (if provided).

(D) Finding p-value using t (click me)

(1) go to t table

(2) find df row in problem: df = n - 1

(3) locate the value of t* on that row

(4) Note: probabilities are given by the subscript in t a at top of columns

A p-value calculated using a t table many times results in a range of values instead of a single value. True False (click one)

When a p-value may be calculated using either a Z* or t* (used in a standard t table), which is most likely to given the most accurate answer?

You should now:

Go on to Hypothesis Test: Examples
or
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Please reference "BA501 (your last name) Assignment name and number" in the subject line of either below.

E-mail Dr. James V. Pinto at BA501@mail.cba.nau.edu
or call (928) 523-7356. Use WebMail for attachments.

	Statistical	Decision
Actual Situation	Reject H_o	FTR(support) H_o
H_o True	Type I Error (a)	correct
H_o False	correct	Type II Error (b)