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Lecture - 7Hypothesis Formulation and testing

Continued

Hypothesis testing tools

- Testing for goodness of fit
- Mostly nominal data e.g. counts or frequencies of

items or events of groups but continuous data can

also be tested - 1. Chi-square or ?2 test
- - Karl Pearson (around 1900)
- - Binomial normal distributions
- - Discrete or continuous series
- 2. Kolomogorov-Smirnov (K-S) test
- - Normal distribution (Discrete or continuous)

Hypothesis testing tools

- H0 there is no difference between observed (O)

and expected (E) frequencies. - H1 there is a difference
- Chi-square (?2) test
- Example 1 In a plant breeding trial, if you get

78 plants with yellow flowers and 22 green

flowers (ratio 3.51). It was supposed to be 31

ratio, test it whether they are statistically

different or not.

Hypothesis testing tools

?2 test ?2 ? (O-E)2/E (78-75)2 /75

(25-22)2 / 25 32/7532/25 0.120.36

0.48 d.f. No. of classes 1 2-11 P0.49

which is 0.5 gt p gt 0.25

Therefore, Accept H0 which means the ratio of

yellow and green flowers is statistically similar

to the ratio found by Mendel i.e. 31

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Hypothesis testing tools

Example 2

- This comes from
- 9331 ratio.
- Y- Yellow, W-wrinkled
- S smooth G-green
- Here,
- d.f. 4-1 3
- 2 0.05,3 7.815 and ? 2 0.01,3 11.34 ,

Plt0.05, - Therefore, Reject Ho, means the ratios are

different

Hypothesis testing tools

Example 3 Application

Test whether a coaching class (intervention

program) has impacts on Students performance or

not?

,

,

,

,

Hypothesis testing tools

Example 3 contd..

22.01.

d.f. (r-1) (c-1) (2-1) ( 2-1) 1 ?2 0.01, 1

6.63 (from table) Plt0.05, therefore, reject Ho

which means the coaching (intervention program)

has positive impacts or has helped graduate more

students and should be recommended.

Hypothesis testing tools

Example 4 Continuous series

Fit the normal curve from the observed frequencies

use the following formula to find the

standardized frequencies

No. of fish (freq)

Fish size (g)

Hypothesis testing tools

Example 4 Continuous series e.g. test whether

size distribution of fish at harvest is normal

Fish size (g)

Hypothesis testing tools

Example 4 Continuous series e.g. test whether

size distribution of fish at harvest is normal

Fish size (g)

2. Kolmogorov-Smirnov (K-S) test - for discrete

- Considers the cumulative frequency
- More expanded form compared to Chi-square test

(frequency classes or category) - D max maximum difference in cumulative freq.

i.e. / d / - Example
- A total of 35 dogs were tested individually which

feed they prefer one by one (not in a group in

which action of one may affect others) - H0 Moisture content of diet do not affect

preference by dog - HA Dog prefer diets with high moisture content

2. Kolmogorov-Smirnov (K-S) test - for discrete

- D max max. difference in cumulative freq. i.e.

/d/ 9 - No. of classes (k) 5 and total no. of data (n)

35 - Dmax, 0.05, 5, 35 7, reject H0

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Hypothesis testing tools

2. Kolmogorov-Smirnov (K-S) test (For

continuous data)

Chi-square test

K-S test

2. Kolmogorov-Smirnov (K-S) test continuous

series (grouped data) H0 Distribution of moth in

a tree trunk is not different (or the

distribution is homogenous). HA More moths are

found lower parts of the tree trunk

- D max max. difference in cumulative freq. i.e.

/d/ 4 - No. of classes (k) 5 and total no. of data (n)

15 - Dmax, 0.05, 5, 15 5, Do not reject H0 ie.

Distribution of moths on the tree trunk is

similar (not heterogeneous)

Comparison of two samples

No sig. difference No sig.

difference Sig. different (overlap is lt5)

Comparison of two means

- Population mean (?) and sample mean (x)
- Hypothetical mean and sample mean
- Two sample means (e.g. Sample A B)

Z - test

- Comparison between two means
- Population mean (?) and sample means (x )
- Use population standard deviation
- Sample size should be more than 30 so that the

distribution need not be normal

Z ( x - µ ) / s vn

Students t - tests

Student" (real name W. S. Gossett

1876-1937) - developed the method.

- Comparison between two means
- Population mean (?) and sample means (x)
- One treatment sample vs hypothetical or standard

value - Two treatment means
- Sample size can be lt30 but they have to have

normal distribution - Use sample Standard Deviation

Students t - tests

- Types
- One-sample t-test
- Paired/matched sample t-test
- Independent sample t-test
- One-sample or single sample t-test
- Comparison of a sample mean against a known

standard or known number (often 0). - Example
- H0 Tilapia do not grow in cold season lt20C (0

g/fish/day) - HA Tilapia grow even below 20C (gt0 g/fish/day)
- A trial was conducted using 4 ponds stocked with

1 fish/m2, we found 1.0 g DWG and 0.1 SD

- One-sample or single sample t-test
- H0 Test the hypothesis whether tilapia really

grew or not or the 1.0 g is significantly higher

than 0 or not? - Here, µ 0, x 1.0g, s 0.1g

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Two-sample t-test

- Calculate the pooled variance
- As the sample size can be different, pooled

variance is calculated taking weighted average of

the variance. - Population pooled Variance s12 /n1 s22 /n2
- Sample Pooled Variance S12/n1 S2 2 /n2
- Degree of freedom n1 n2 -2
- Calculate test statistics and compare with the

tabulated value for P

Example Rats injected with a drug to increase

pain sensitivity from a heat source in 4.3

seconds, The mean difference of 2.8 seconds is

statistically significant (t(12) 2.33, p

.038). Thus, the drug indeed increased pain

sensitivity.

Control 2.6 10 9.5 7.4 6.9 8.5 5.2

Drug 2.2 3.8 7.1 2.7 6.2 5.3

3.1

MS Excel Results

Survey research on Aquaculture Productivity

- Suppose, a student gathered data from 35 farmers

in two districts using survey method from to - - compare productivity between districts A B
- compare District A and B separately against the

national average.

District A District B

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Results

Paired-sample t-test

- To compare means on the same or related subject

over time or in differing circumstances - t Mean difference between pairs / SE of mean

difference - d / svn
- N no. of pairs

Data set 1 Data set 2

Pond A Pond B

1 2 3 4 5 6

Sampling points (weeks - pairs)

Paired-sample t-test

- Note Another aspect of these type of data is to

see the correlation - - There can be both or only one
- A B correlated (may or may not differ)
- A C No correlation, sig. difference
- B C No correlation, no difference

Paired-sample t-test

- Example for paired t-test try in the lab

session.

Lab session after 15 minutes

Statistical principles and t-tests http//www.ph

ysics.csbsju.edu/stats/t-test.html http//www.stat

soft.com/textbook/stathome.html

Thank you!

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