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T TEST. DEFINITION OF TERMS. T – Test – the statistical test for comparing a mean with a norm or for comparing two means with small sample sizes (n≤30). Formula: __ X - µ

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Definition of terms
DEFINITION OF TERMS

  • T – Test – the statistical test for comparing a

    mean with a norm or for comparing

    two means with small sample sizes

    (n≤30).

    Formula:

    __

    X - µ

    t = ———

    SD/√n


T test

_

  • Mean ( X ) – the most common measure of central

    tendency, denoted by µ in the

    population and by in the sample.

    Formula: _

    X = ΣX

    n

    Σ – summation or to add

    µ - population mean

    X – individual observations

    n – number of observations


T test

  • Population – the entire collection of observations

    or subjects that have something in

    common and to which conclusions are

    inferred.

  • Standard Deviation – the most common measure of

    dispersion or spread, denoted by

    σ in the population and SD or s

    in the sample. It can be used with

    mean to describe the distribution

    of observations.


T test

Formula:

_

SD = √Σ (X – X)²

n – 1

Hypothesis Test – an approach to statistical inference

resulting in a decision to reject or

not to reject the null hypothesis.


Steps in hypothesis testing
STEPS IN HYPOTHESIS TESTING

STEP 1: State the research question in terms of

statistical hypotheses or formulate the hypothesis

Null Hypothesis (Ho) – a statement claiming that there is

no difference between the

assumed or hypothesized value

and the population mean.

Alternative Hypothesis (Hı)- a statement that disagrees

with the null hypothesis.


Step 2 decide on the appropriate test statistic

Step 2: Decide on the appropriate test statistic.

__

X - µ

t = ———

SD/√n


Step 3 select the level of significance for the statistical test
STEP 3: Select the level of significance for the statistical test.

Level of Significance – the probability of incorrectly

rejecting the null hypothesis in a

test of hypothesis, denoted by α

when it is actually true (and

concluding there is difference

when there is nor.

- traditional values used for α are

0.05, 0.01 and 0.001.


T test

STEP 4: Determine the value the test statistic must attain to be declared significant. Determine the rejection region or the critical region.

Rejection Region – is a set of values of the test statistic for

which the null hypothesis is rejected

in a hypothesis test.

Acceptance Region – non-critical region

Critical Value – the value that a test statistic must exceed

for the null hypothesis to be rejected.

- refer to the critical values for t

distribution table.


T test

Degrees of Freedom (df) – the precise size of the standard

deviation depending on a

complex concept related to

the sample size which is

related to the number of times

sample information is used.

- n -1 (df)

STEP 5: Perform the calculation.

STEP 6: Interpret and state the conclusion.


T test

EXAMPLE: YLO Company, manufacturer of Yummy

Oats is claiming that their product contains

8 grams of fiber per 35 gram-pack. VIN

Company, its direct competitor conducts an

independent test of the product. They test

15 samples of Yummy Oats and it found out

to have 8.00, 8.25, 8.16, 7.92, 7.81, 7.93,

8.01, 8.02, 8.50, 8.16, 7.71, 7.81, 7.62, 7.81

and 7.86 of fiber content. If 0.01 level of

significance is appropriate, would VIN

Company agree on the claim of YLO Co.?


T test

STEP 1: H o: µ = 8 grams : Yummy Oats contains 8 grams of fiber/pack Hı: µ ≠ 8 grams : Yummy Oats does not contain 8 grams of fiber/pk

STEP 2: __

X X-X (X-X)²

8.00 0.03 0.0009

8.25 0.28 0.0784

8.16 0.19 0.0361

7.92 0.05 0.0025

7.81 0.16 0.0256

7.93 0.04 0.0016

8.01 0.04 0.0016

8.02 0.05 0.0025

8.50 0.53 0.2809

8.16 0.19 0.0361


T test

7.71 0.26 0.0676 7.81 0.16 0.0256 7.62 0.35 0.1225 7.81 0.16 0.0256 7.86 0.11 0.0121

_

X = 119.57

15

= 7.97 Σ = 0.7196


Sd 0 7196 15 1 0 7196 14 0 0514 0 2267
SD =√0.7196/15-1 = √0.7196/14 = √0.0514 = 0.2267

7.97 – 8 -0.03 -0.03

T = ÷ = ÷ = ÷

0.2267/√15 0.2267/3.8729 0.0585

T = -0.5128


Step 3 level of significance 0 01
STEP 3: Level of Significance: 0.01

STEP 4:

Degrees of Freedom: 15 – 1 = 14

Critical Value : -2.624

RR AR

__________________________________

-2.624 -0.5128 0


T test

STEP 6: The computed value of t is -0.5128 is at

the acceptance region.

Accepts Null Hypothesis

CONCLUSION:

VIN Company would agree to the claim

of YLO Company that Yummy Oats has

8 grams per 35 gram/pack of fiber

content.


T test

EXAMPLE: It was indicated in the label of Vita C

that it contains 235mg of ascorbic

acid. An independent test was done

to prove whether the claim of the

manufacturer is true or not. Ten samples

were tested and the results are as follows:

230, 232, 228, 231, 218, 225, 221, 220, 223,

and 228. If 0.01 level of significance is

appropriate, determine whether to accept or

reject the claim of Vita C manufacturer.


T test

STEP 1: H o: µ = 235 mg : Vita C contains 235 mg ascorbic acid Hı: µ ≠ 235 mg : Vita C does not contain 235 ascorbic a.

__

X X-X (X-X)²

230 4.4 19.36

232 6.4 40.96

228 2.4 5.76

231 5.4 29.16

218 7.6 57.76

225 0.6 0.36

221 4.6 21.16

220 5.6 31.36

223 2.6 6.76

228 2.4 5.76


X 2256 10 225 6 218 4
_ X = 2256/10 = 225.6 Σ = 218.4

SD = √218.4/10-1 = √218.4/9 = √24.27 = 4.93

T = 225.6-235/4.93/√10 = -9.4/4.93/3.16 = -9.4/1.56

= -6.03

STEP 3: Level of Significance: 0.01

STEP 4: Degrees of Freedom: 10-1 = 9


Rr ar 6 03 2 821 0
RR AR _______________________________________ -6.03 -2.821 0

STEP 6: The computed value of t is -6.03 is at the

rejected region.

Rejects Null Hypothesis.

CONCLUSION: The independent test for Vita C

containing 235 mg of ascorbic

acid is rejected.