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Null Hypothesis:$H_0:{渭}_1鈭�{渭}_2=0$

Alternate Hypothesis:$H_a:{渭}_1鈭�{渭}_2<0$

The critical value for a left-tailed test$伪=0.1$ is$-1.28$ .

If $z_{data}<鈭�1.28$then, reject the null hypothesis.

$\begin{array}{l}z=\frac{\stackrel{炉}{d}鈭�0}{\sqrt{\left(\frac{{s_d}^2}{n_d}\right)}}\\ =\frac{鈭�3.5鈭�0}{\sqrt{\left(\frac{21}{38}\right)}}\\ =\frac{鈭�3.5}{0.7434}\\ =鈭�4.71\end{array}$

Therefore, the value of paired difference test is $-4.71$.

The null hypothesis will be rejected, and it can be concluded that $H_a:{渭}_1鈭�{渭}_2<0$.

1. By Central Limit Theorem, the sampling distribution$\stackrel{炉}{d}$ follows a normal distribution.
2. The sample differences are randomly selected from the population differences.
3. The sample size of the no. of pairs is large, which is more than $30$.

Confidence interval $=90\%$

Level of Significance $\left(伪\right)=0.10$

Confidence Interval $=\stackrel{炉}{d}卤z_{伪/2}\left(\sqrt{\frac{{s_d}^2}{n_d}}\right)$

$\begin{array}{l}=鈭�3.5卤z_{0.05}\left(\sqrt{\frac{21}{38}}\right)\\ =鈭�3.5卤1.645\left(0.7434\right)\\ =鈭�3.5卤1.223\\ =(鈭�4.723,鈭�2.277)\end{array}$

Therefore,$90\%$the confidence interval for the mean difference ${渭}_d$is $(鈭�4.723,鈭�2.277)$.

The confidence intervals tell us about the difference between population means of an interval for all possible values. So, it gives more information than the z-test.