## Test of significance of singal mean

[mathjax] Z- Test Singal Mean Test Setting
Null Hypothessis (H_{0}: mu=mu_{0}) (Given value of (mu) )
There are no significant different between sample mean and population mean
Alternative Hypothessis (H_{1}: mu neq mu_{0}) (Two tail test)
There are significant different between sample mean and population mean (H_{1}: mu>mu_{0}) (Right tail test or one tail test)
Population mean is greater than Sample mean
(H_{1}: mu<mu_{0}) (Left tail test or one tail test)
Population mean is less than Sample mean
Test of statistics,
(Z_{c a l}=frac{bar{X}-mu}{frac{sigma}{sqrt{n}}}=frac{bar{X}-mu}{frac{s}{sqrt{n}}}(hat{sigma}=) s for largr sample ())
Find (Z_{text {tab }}) from table using significance level (1 %, 2 %), or (5 %). If significance level
Is not given then use (5 %)
Coparission and Dissision
Since (left|Z_{c a l}right| leqleft|Z_{t a b}right|) it is not significant and (mathrm{H}_{0}) accepted, (mathrm{H}_{1}) rejected means
The sample mean is same as the Population mean
Since (left|Z_{c a l}right|>left|Z_{t a b}right|) it is significant and (H_{0}) rejected, (H_{1}) accepted means Sample mean is same as Population mean

## Z- Test Double Mean Test Set up

Null Hypothessis (H_{0}: mu_{1}=mu_{2}) (Given value of (mu) )
There are no significant different between two population mean
Alternative Hypothessis (H_{1}: mu_{1} neq mu_{2}) (Two tail test)
There are significant different between two population mean
(H_{1}: mu_{1}>mu_{2}) (Right tail test or one tail test)
First population mean is greater than second one
(H_{1}: mu_{1}<mu_{2}) (Left tail test or one tail test)
First population mean is less than second one
Test of statistics,
(Z_{c a l}=frac{bar{X}_{1}-bar{X}_{2}}{sqrt{frac{sigma_{1}^{2}}{n_{1}}+frac{sigma_{1}^{2}}{n_{2}}}}=frac{bar{X}_{1}-bar{X}_{2}}{sqrt{frac{s_{1}^{2}}{n_{1}}+frac{s_{1}^{2}}{n_{2}}}}left(right.) for large sample (hat{sigma}_{1}=s_{1}) and (left.hat{sigma}_{1}=s_{2}right))
Find (Z_{t a b}) from table using significance level (1 %, 2 %), or (5 %). If significance level is not given then use (5 %)
Coparission and Dissision
Since (left|Z_{c a l}right| leqleft|Z_{t a b}right|)
it is not significant and (mathrm{H}_{0}) accepted, (mathrm{H}_{1}) rejected means There are no significant different between two population mean
Since (left|Z_{c a l}right| leqleft|Z_{t a b}right|)
it is significant and (H_{0}) rejected, (H_{1}) accepted means
There are no significant different between two population mean

## Test of significance for single proportions

Step 1
(H_{0}: P=P_{0}). That is, population proportion has some specified value (P_{0}).
(H_{1}: P neq P_{0}) (Two tailed test). That is, population proportion is not equal to (P_{0}).
(mathrm{H}_{1}: mathrm{P}>mathrm{P}_{0}) (Right tailed test). That is, the population proportion is greater than (mathrm{P}_{0}).
(mathrm{H}_{1}: mathrm{P}<mathrm{P}_{0}) (Left tailed test). That is, the population proportion is less than (mathrm{P}_{0}).
It is noted that one should be chosen only one alternative hypothesis depending upon the nature of the problem.
Step – 2
Test Statistics: Under (mathrm{H}_{0 mathrm{Z}}=frac{p-P}{sqrt{frac{P Q}{n}}})
Where, (P=) population proportion of success
(mathrm{p}=) sample proportion of success (=frac{X}{n})
(X=) number of successes
(mathrm{n}=) sample size or no. of trials
SE. ((P)=) Standard error of proportion (=sqrt{frac{P Q}{n}})
Step – 3
Write down the tabulated value of (Z) at (alpha) level of significance according as whether the alternative hypothesis is one tailed test or two tailed tests.
Step-4
Conclusion:
(left|Z_{text {cal }}right| leqleft|Z_{text {tab }}right| rightarrow mathrm{H}_{0}) is accepted (rightarrow) it is not significant
(left|Z_{c a l}right|>left|Z_{t a b}right| rightarrow H_{1}) is accepted (rightarrow) it is not significant

## Test of Significance for the Difference of Two Proportions

Test of Significance for Difference of Two Proportions
Set up the null hypothesis and alternative hypothesis
Step 1
Null hypothesis (mathrm{H}_{0}: mathrm{P}_{1}=mathrm{P}_{2}=mathrm{P}) (say). That is, there is no significant difference between two sample proportions (p_{1}) and (p_{2}).
Alternative hypothesis (H_{1}: P_{1} neq P_{1}) (Two tailed test). That is, there is significant difference between two sample proportions (p_{1}) and (p_{2}).
(H_{1}: P_{1}>P_{1}) (Right tailed test). That is, one group population proportion is greater than other group population proportion.
(H_{1}: P_{1}<P_{2}) (Left tailed test). That is, one group population proportion is less than other group population proportion.
Step 2
Test statistic: Under (mathrm{H}_{0}: mathrm{P}_{1}=mathrm{P}_{2}=hat{P}_{text {, then test statistic is }}^{text {, }}) is
(Z_{text {cal }}=frac{P_{1}-P_{2}}{sqrt{P Qleft(frac{1}{n_{1}}+frac{1}{n_{2}}right)}})
where (P=) the common population proportion under (mathrm{H}_{0}) is unknown and we use its unbiased estimate provided by both samples taken together which is given by (hat{P}=frac{X_{1}+X_{2}}{n_{1}+n_{2}}=frac{n_{1} p_{1}+n_{2} p_{2}}{n_{2}+n_{2}})
And (hat{Q}=1-hat{P})
If (P) unknown then test statistics is
(Z_{mathrm{cal}}=frac{p_{1}-p_{2}}{sqrt{hat{P} hat{Q}left(frac{1}{n_{1}}+frac{1}{n_{2}}right)}})
Step 3
Obtain tabulated (critical) value of Z at (alpha) level of significance for appropriate alternative hypothesis.
The most commonly used is (alpha=5 %).
Step 4
Decision: Make a decision by comparing the calculated value of (Z) with critical value of (Z).
(left|Z_{text {cal }}right| leqleft|Z_{t a b}right| rightarrow H_{0}) is accepted (rightarrow) it is not significant
(left|Z_{text {cal }}right|>left|Z_{text {tab }}right| rightarrow H_{1}) is accepted (rightarrow) it is not significant.

## Test of significance of a Single Mean

(mathrm{H}_{0}: mu=mu 0). That is, the population mean has some specified value (mu_{0})
(mu neq mu_{0}) (Two tailed test). That is, the population mean is not equal to (mu_{0}).
(mathrm{H}_{1}: mu>mu_{0}) (Right tailed test). That is, the population mean is greater than (mu_{0}).
(mathrm{H}_{1}: mu<mu_{0}) (Left tailed test). That is, the population mean is less than (mu_{0}).
(mathrm{t}_{mathrm{cal}}=frac{bar{X}-mu}{frac{s}{sqrt{n-1}}}) (when we are given the sample s.d. (=mathrm{s}) )
(mathrm{t}_{text {cal }}=frac{X-mu}{frac{S}{sqrt{n}}}) (when we are given the sample data)
(mathbf{S}^{2}=frac{1}{n-1} boldsymbol{D}(boldsymbol{X}-bar{X})^{2})
(S^{2}=frac{1}{n-1}left[Sigma X 2-frac{(Sigma X)^{2}}{n}right])
(S^{2}=frac{1}{n-1}left[Sigma d 2-frac{(Sigma d)^{2}}{n}right])
Test of Significance for Difference between two Independent Means
(mathrm{H}_{0}: mu 1=mu 2). That is, the samples have been drawn from the normal populations with the same means.
(mathrm{H}_{1}: mu_{1} neq mu_{2}) (Two tailed test). That is, the samples have not been drawn from the normal populations with the same means.
(mathrm{H}_{1}: mu_{1}>mu_{2}) (Right tailed test). That is, mean of first population is higher than the mean of the second population.
(mathrm{H}_{1}: mu_{1}<mu_{2}) (Left tailed test). That is, mean of first population lower than mean of second population
begin{equation*}
begin{aligned}
&mathrm{t}_{mathrm{cal}}=frac{bar{X}-bar{X}}{sqrt{S^{2}left(frac{1}{n_{1}}+frac{1}{n_{1}}right)}} \
&mathrm{S}^{2}=frac{1}{n 1+n 2-2}left[left(Sigma X 1-bar{X}_{1}right)^{2}+left(Sigma X 2-bar{X}_{2}right)^{2}right] \
&mathrm{S}^{2}=frac{1}{n 1+n 2-2}left[Sigma X_{1}^{2}-frac{(Sigma X 1)^{2}}{n 1}+Sigma X_{2}^{2}-frac{(Sigma X 2)^{2}}{n 2}right] \
&mathrm{S}^{2}=frac{1}{n 1+n 2-2}left[Sigma d_{1}^{2}-frac{(Sigma d 1)^{2}}{n 1}+Sigma d_{2}^{2}-frac{(Sigma d)^{2}}{n 2}right] \
&mathrm{S}^{2}=frac{n_{1} s_{1}^{2}+n_{2} s_{1}^{2}}{n_{1}+n_{2}-2}
end{aligned}
end{equation*}

## Paired t-test for Difference of Means

(mathrm{H}_{0}: mu x=mu y). That is, there is no significant difference in the observations before and after treatment. (mathrm{H}_{1}: mu x neq mu y). That is, there is a significant difference in the observations before and after treatment.
(mathrm{H}_{1}: mu x>mu y), (Right tailed test). That is, the treatment is not effective.
(mathrm{H}_{1}: mu x<mu y) (Left tailed test). That is, the treatment is effective.
begin{equation*}
mathrm{t}_{mathrm{cal}}=frac{bar{d}}{sqrt{frac{s^{2}}{n}}} sim mathrm{t}_{mathrm{n}-1}
end{equation*}
where (mathrm{d}=mathrm{X}-mathrm{Y}=) difference between two set of observations
(bar{d}=) Mean of the difference and
begin{equation*}
mathrm{S}^{2}=frac{1}{n-1}left[Sigma d^{2}-frac{(Sigma d)^{2}}{n}right]
end{equation*}
t-test for Significance of an Observed Sample Correlation Coefficient
(mathrm{H}_{0}: rho=0). That is, population correlation coefficient is zero.
Alternative hypothesis (mathrm{H}_{1}: rho neq 0) That is, population correlation coefficient is not zero.
(mathrm{H}_{1} equiv rho>0) (Right tailed test). That is, there is positively correlated in the population.
(underline{underline{mathrm{H},:}} rho<0) (Left tailed test). That is, there is negatively correlated in the population.
(underline{mathrm{t}_{mathrm{cal}}=frac{r}{sqrt{1-r^{2}}}} times sqrt{n-2})
Conclusion: Make a decision by comparing the calculated value oft with tabulated value of (t)
If (left|mathrm{t}_{text {cal }}right| leqleft|mathrm{t}_{text {tab }}right|), it is not significant and (mathrm{H}_{0}) is accepted.
If (left|t_{text {caal }}right|>left|t_{text {tab }}right|), it is significant and (H_{1}) is accepted.
Confidence Limits in Estimating Population Mean ((mu)) for Small Samples
Confidence Interval
(bar{X} pm mathrm{t} alpha_{/ 2}(mathrm{n}-1) times underline{underline{mathrm{SE}}(bar{X})})
(underline{underline{operatorname{SE}}(bar{X}})=frac{S}{sqrt{n}})
(=frac{s}{sqrt{n-1}}) (if sample standers deviation given)
If data is given
begin{equation*}
begin{aligned}
&mathbf{S}^{2}=frac{1}{n-1}left[Sigma d 2-frac{(Sigma d)^{2}}{n}right] \
&S^{2}=frac{1}{n-1} Sigma(X-bar{X})^{2} \
&S^{2}=frac{1}{n-1}left[Sigma X 2-frac{(Sigma X)^{2}}{n}right]
end{aligned}
end{equation*}