W5 One Sample T Test.pdfw5 One Sample T Test.pdf

Minggu 5

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Nyatakan H0 dan Ha

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Pengujian Hipotesis

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Nilai Kritikal

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Nilai Statistik Pengujian • Ini adalah nilai yang dikira dan dijadikan bukti sama ada hipotesis sifar benar atau salah. • Jika nilai statistik pengujian masuk dalam kawasan kritikal maka H0 adalah salah, ditolak dan Ha gagal ditolak. • Jika nilai statistik pengujian masuk dalam kawasan tidak kritikal maka H0 adalah benar, maka gagal menolak H0.

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TABURAN PERSAMPELAN

Hipotesis tak berarah

Hipotesis berarah +ve

Hipotesis berarah -ve

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NILAI-NILAI KRITIKAL BAGI UJIAN-Z UJIAN

0.01 1% TIDAK BERARAH ±2.58

0.05 5% ±1.96

BERARAH POSITIF BERARAH NEGATIF

+2.33

+1.64

-2.33

-1.64 7

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LANGKAH PENGUJIAN HIPOTESIS

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Statistik Pengujian

Z kira =

t kira

= 10

Ujian-Z dan Ujian-t (membanding satu kumpulan dengan “norms”)

— Digunakan bagi skor-skor yang bertabur secara normal (juga dipanggil skor-skor piawai – skor z) dan dengan itu pengujian hipotesis ini dinamakan ujian-z

Z kritikal

=

—I — α digunakan untuk membanding sesuatu kumpulan dengan “norm” bagi sesuatu populasi. Bilangan sampel bagi penggunaan ujian-z lazimnya adalah lebih besar daripada 30. 11

DISEBALIKNYA, jika taburan adalah Ujian-z dan skor-skor Ujian-t normal tetapi bilangan yang digunakan (membanding satusampel kumpulan dengan “norms”) adalah kecil (n<30) maka UJIAN-t digunakan pakai dan statistik pengujiannya berubah menjadi………..

t kritikal

=

oleh G.W. Gossett di bawah nama samaran ia itu, Student t, dengan itu ujian tersebut dinamakan ujiant.

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One Sample Z-test

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One Sample Z-test • If the calculated Z falls outside of the “likely” zone of our distribution, we reject the null hypothesis. • The area in the distribution that falls outside of the “likely” zone is called the “region of rejection” the probability of making a Type 1 error • Typically a is set at .05

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A sampling distribution for H0 showing the region of rejection for α = .05 in a 2-tailed z-test.

2-tailed regions

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A sampling distribution for H0 showing the region of rejection for α = .05 in a 1-tailed z-test.

1-tailed region, above mean

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A sampling distribution for H0 showing the region of rejection for α = .05 in a 1-tailed z-test where a decrease in the mean is predicted.

1-tailed region, below mean

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Contoh 1 Seorang pensyarah telah membuat rekod markah pencapaian pelajar untuk subjek KP2 dan didapati min populasi adalah μ, is 72. Kumpulan 36 orang pelajar yang mendaftar untuk semester berikutnya mempunyai pencapaian dengan min markah ialah 75.2. Apakah terdapat perbezaan pencapaian pelajar semester tersebut jika dibandingkan dengan pencapaian keseluruhan populasi.

Use α = 0. 05 and σ = 12

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Use α = 0. 05 and σ = 12

Hipotesis Nol

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H0 :Tidak terdapat perbezaan pencapaian antara sampel pelajar dengan populasi

Ha: Terdapat perbezaan pencapaian antara sampel pelajar dengan populasi

TEST STATISTIC Z *

z =

X − µ0

σ n

75 .2 − 72 = = 1.60 12 36

TEST CRITERIA α = 0.05

zα = z0.05 = ________

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Kawasan penolakan dan p-value

0

1.60

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Kesimpulan Keputusan: ?

Computing Confidence Interval • We seek the highest and lowest µ that are not significantly different from the sample mean. • You are calculating interval that the μ of the population that you are studying will fall on AND you are K% sure of this interval.

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Confidence Interval Estimation of Population Mean, μ, when σ is known Assumptions – Population standard deviation σ is known – Population is normally distributed – If population is not normal, use large sample

(where Z is the normal distribution’s critical value for a probability of α/2 in each tail)

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• Consider a 95% confidence interval: 1 − α = .95

α = .05

.475

α = .025 2

α / 2 = .025

.475

α = .025 2 Z

Z= -1.96 Lower Confidence Limit

0 Point Estimate

Z= 1.96 Upper Confidence Limit

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Two-tailed hypothesis • If we take a sample of 30 students and find their average IQ to be 106.58, we can test whether the population from which our sample came had an average IQ of 100 H0: m=100 (average IQ) Ha:m ≠ 100 • Convert our sample mean to Z statistics (106.58-100)/2.74 = 2.4

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Z=-2.4

Z= 2.4 29

The t Distribution We use t when the population variance is unknown (the usual case) and sample size is small. If you use a stat package for testing hypotheses about means, you will use t.

The t distribution is a short, fat relative of the normal. The shape of t depends on its degree of freedom (n-1). As N becomes infinitely large, t becomes normal.

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One-Sample T-test

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One-Sample T-test • After calculating t-statistic, we find associated p-value • SPSS does this for us • T is about the same as Z in large samples. • For this class we will always use T

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Making a decision

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One sample t-test As sample size increases, the results from a t-test approximate a z-test SPSS only does one sample t-tests 35

Formula for one sample t -test

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Confidence interval for one sample t test

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Summary : Confidence Interval (1 - a)% confidence interval for a population parameter P ( C. I. encloses true population parameter ) = 1 - a Note: a= P(Confidence Interval misses true population parameter ) “Proportion of times such a CI misses the population parameter”

Margin of Error Point estimate

±

sample statistic ex:

X

critical value

·

or

zα / 2

tα / 2

Std. dev. of point estimate

standard deviation of sampling distribution (aka “Standard Error”)

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Contoh

ID

Skor

ID

Skor

1

83

11

127

2

122

12

115

3

126

13

98

4

110

14

136

5

114

15

87

6

129

16

138

7

106

17

84

8

82

18

93

9

112

19

111

10

131

20

148

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Latihan Purata skor Matematik pada peperiksaan yang lepas ialah 75.5. Seramai 200 orang pelajar mengambil peperiksaan tersebut min skor Matematik ialah 78 dan sisihan piawai ialah 6.7. Adakah terdapat perbezaan skor prestasi skor antara populasi dengan sampel pada aras signifikan 0.05?

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