Sample size n = 100,
Sample mean \(\\ \bar { X } =3.5\)
Population mean μ = 4
Population standard deviation σ = 3
Null Hypotheses: There is no significant difference in the mean. i.e., H_o : μ = 4
Alternative Hypotheses : There is Significant difference in the mean.
i.e., H₁ : μ ≠ H
The level of significance ∝ = 5% = 0.05
Applying the test statistic,\(Z=\frac { \bar { X } -\mu }{ \frac { \sigma }{ \sqrt { n } } } \)
\(\Rightarrow Z=\frac { 3.5-4 }{ \frac { 3 }{ \sqrt { 100 } } } =\frac { -.5 }{ .3 } =-1.667\)
\(\Rightarrow |Z|=1.667\)
\({ Z }_{ \frac { \alpha }{ 2 } }=1.96\)
Here Z < \({ Z }_{ \frac { \alpha }{ 2 } }\)i.e., 1.667<1.96
Inference: Since Z<\({ Z }_{ \frac { \alpha }{ 2 } }\)at 5% level of significance, the null hypothesis H₀ is accepted. Hence there is no Significant difference in the mean.

\( \text { Given } n=1000, \bar{X}=119, \sigma=30\)
\(S . E=\sigma / \sqrt{n}\)
\(=30 / \sqrt{1000} \)
\(=30 / 31.623=0.9487 \)

There are many ways to select a sample of 10 3-digit even numbers. From the table, start from the first number and move along the column. Select the first three digits as the number. If it is an odd number, move to the next number. The selected sample is 416, 664, 952, 748, 524, 914, 154, 340, 140, 276.
\(\begin{array}{|l|l|l|l|l|l|l|l|} \hline 2952 & \underline{6641} & 3992 & 9792 & 7969 & 5911 & 3170 & 5624 \\ \hline \mathbf{4 1 6 7} & \underline{9524} & \mathbf{1 5 4 5} & 1396 & 7203 & 5356 & 1300 & 2693 \\ \hline 2670 & \underline{7483} & \underline{3408} & \underline{2762} & 3563 & 1089 & 6913 & 7991 \\ \hline 0560 & \underline{5246} & 1112 & 6107 & 6008 & 8125 & 4233 & 8776 \\ \hline 2754 & \underline{9143} & \underline{1405} & 9025 & 7002 & 6111 & 8816 & 6446 \\ \hline \end{array}\)

Given \(\sigma\) = 10, S.E. \(\bar { X } \) = 3
We know that S.E = \(\frac { \sigma }{ \sqrt { n } } \)
Therefore, \(3=\frac { 10 }{ \sqrt { n } } \Rightarrow \sqrt { n } =\frac { 10 }{ 3 } \)
Taking Squaring on both sides we get
\(n=\left(\frac{10}{3}\right)^{2}=\frac{100}{9}=11.11 \cong 11\), 
The required sample size is 11.

Merits :
(a) A random stratified sample is superior to a simple random sample because it ensures representation of all groups and thus it is more representative of the population which is being sampled.
(b) A stratified random sample can be kept small in size without losing its accuracy.
(c) It is easy to administer, if the population under study is sub-divided.
(d) It reduces the time and expenses in dividing the strata into geographical divisions, since the government itself had divided the geographical areas.
Demerits :
(a) To divide the population into homogeneous strata (if not divided), it requires more money, time and statistical experience which is a difficult one.
(b) If proper stratification is not done, the sample will have an effect of bias.
(c) There is always a possibility of faulty classification of strata and hence increases variability.