Overview Formula Exercise – 1 Exercise – 2 Exercise – 3 Exercise – 4 Exercise – 5 TU Solution

Exercise – I

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

Exercise – II

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.

Exercise – III

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*}

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