MEV 019: Unit 11 - Statistical Analysis-II
UNIT 11: STATISTICAL ANALYSIS – II
11.1
Introduction
In Unit 10, we studied large sample hypothesis
tests using the Z-distribution.
In this unit, we focus on small sample tests (n < 30), where
population standard deviation is usually unknown, and we rely on the t-distribution.
We also study chi-square tests for
categorical data analysis and the F-test for comparing variances.
These tests are widely applied in scientific research, including environmental
studies, to verify theoretical models, test relationships, and compare group
variability.
11.2
Objectives
After studying this unit, you will be able to:
- Perform
small sample t-tests for population means.
- Compare
two small-sample means (independent and paired).
- Conduct
chi-square tests for goodness of fit and independence.
- Perform
F-tests for comparing two population variances.
- Interpret
statistical results in practical contexts.
11.3
Procedure for Small Sample Test
Small sample tests are based on the Student’s
t-distribution, developed by W.S. Gosset.
The shape of the t-distribution depends on degrees of freedom (df) and
is broader than the normal curve, especially for small n.
General Steps (similar to large-sample tests):
- State H₀
and H₁.
- Choose α
(level of significance).
- Select
the test statistic.
- Determine
critical t-value from t-table (based on df).
- Compute
test statistic from sample data.
- Compare
with critical value and conclude.
11.3.1 Test for Population Mean
When σ is unknown and n < 30:
t=Xˉ−μs/nt = \frac{\bar{X} - \mu}{s / \sqrt{n}}t=s/nXˉ−μ
Where:
- Xˉ\bar{X}Xˉ
= sample mean
- μ\muμ =
population mean under H₀
- sss =
sample standard deviation
11.3.2 Test for Difference of Two Population Means
(Independent Samples)
t=Xˉ1−Xˉ2Sp1n1+1n2t = \frac{\bar{X}_1 - \bar{X}_2}{S_p \sqrt{\frac{1}{n_1}
+ \frac{1}{n_2}}}t=Spn11+n21Xˉ1−Xˉ2
Where:
Sp=(n1−1)s12+(n2−1)s22n1+n2−2S_p = \sqrt{\frac{(n_1 - 1)s_1^2 + (n_2 -
1)s_2^2}{n_1 + n_2 - 2}}Sp=n1+n2−2(n1−1)s12+(n2−1)s22
- SpS_pSp
= pooled standard deviation
- df =
n1+n2−2n_1 + n_2 - 2n1+n2−2
11.3.3 Paired t-test
Used when data are in pairs (e.g., before-and-after
measurements on same subjects).
Steps:
- Calculate
the differences di=X1i−X2id_i = X_{1i} - X_{2i}di=X1i−X2i.
- Find
dˉ\bar{d}dˉ (mean difference) and sds_dsd (standard deviation of
differences).
- Apply:
t=dˉsd/nt = \frac{\bar{d}}{s_d / \sqrt{n}}t=sd/ndˉ
Where nnn = number of pairs.
11.4
Chi-Square Test ( χ2\chi^2χ2 )
The chi-square test compares observed and expected
frequencies.
11.4.1 Test for Goodness of Fit
Used to check whether sample data match a
theoretical distribution.
χ2=∑(Oi−Ei)2Ei\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}χ2=∑Ei(Oi−Ei)2
Where:
- OiO_iOi
= observed frequency
- EiE_iEi
= expected frequency
- df =
number of categories – 1 – number of estimated parameters
11.4.2 Test for Independence of Attributes
Checks whether two categorical variables are
independent (e.g., gender vs. preference).
Steps:
- Arrange
data in a contingency table.
- Calculate
expected frequencies:
Eij=(Row total)(Column total)Grand totalE_{ij} =
\frac{(\text{Row total})(\text{Column total})}{\text{Grand
total}}Eij=Grand total(Row total)(Column total)
- Apply
the χ2\chi^2χ2 formula and compare with table value at given df.
11.5 F-Test
The F-test compares two sample variances to
check if the populations have equal variances.
11.5.1 Test for Equality of Two Variances
F=s12s22F = \frac{s_1^2}{s_2^2}F=s22s12
Where:
- s12s_1^2s12
= larger sample variance
- s22s_2^2s22
= smaller sample variance
- df₁ =
n1−1n_1 - 1n1−1, df₂ = n2−1n_2 - 1n2−1
Compare calculated F with table value from the
F-distribution.
11.6 Let Us
Sum Up
- Small
sample tests use t-distribution when σ is unknown.
- Independent
and paired t-tests are used for mean comparisons.
- Chi-square
tests handle categorical data analysis (goodness of fit and independence).
- The
F-test checks variance equality between two populations.
These tools are fundamental for validating research
hypotheses, especially with small datasets.
11.7 Key
Words
- t-distribution –
Probability distribution used for small samples.
- Paired
t-test – Test for differences in matched data.
- Chi-square
test – Test for categorical data comparisons.
- Goodness
of Fit – How well observed data match a theoretical
model.
- Independence
of Attributes – No association between two categorical
variables.
- F-test –
Compares two variances for equality.
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