10.4 Two column model step by step example of a hypotheses test

We have bivariate, categorical data. We have two independent groups. We have a sample of 200 adults taking medicine A and 200 adults taking medicine B. Our model will be:

$P_A-P_B~$

$\mathrm{N\; (}{P}_A-P_B,\sqrt{{\hat{p}}_{\mathrm{pooled}}\mathrm{(\; 1\; -}{\hat{p}}_{\mathrm{pooled}}\mathrm{)\; (}\frac{1}{n_1}+\frac{1}{n_2})})$

1. Express the hypothesis to be tested in symbolic form.
2. Write a symbolic expression that must be true when the original claim is false.
3. The null hypothesis is the statement which included the equality.
4. The alternative hypothesis is the statement without the equality.

Null hypothesis in words: The null hypothesis is that the true proportion of adults helped by medication A is equal to the true proportion of adults helped by medication B.

Null Hypothesis symbolically: H ₀ : Proportion $p_A=p_B$

Alternative Hypothesis in words: The alternative is that the true proportion of adults helped by medication A is more than the true proportion of adults helped by medication B.

Alternative Hypothesis symbolically: H _a : Proportion $p_A>p_B$

Randomization Condition: The samples are random samples.

Independence Assumption: Since we know that a random sample was taken it is reasonable to assume independence. The likelihood of having the medicine work for one person has no bearing on if it will work for another.

Independent Groups Assumption: It is reasonable to think that the medicine A group and medicine B group recovery rates are independent of each other.

10% Condition: I assume the number of adults in the community where this was done is more than 4000, so the 200 adults in each group is less than 10% of each population.

Success/Failure Condition: Success = no hives Failure = hives (.94)(200) = 188 and (1 - .94)(200) = 12 and (.90)(200) = 180 and (1 - .9)(200) = 20, all values are larger than 10 so the condition is met.

Compute the test statistic:

• ${\hat{p}}_A=\frac{\mathrm{188}}{\mathrm{200}}\mathrm{=\; 0.94;}{n}_A\mathrm{=\; 200;}$
• ${\hat{p}}_B=\frac{\mathrm{180}}{\mathrm{200}}\mathrm{=\; 0.90;}{n}_B\mathrm{=\; 200}$

• ${\hat{p}}_{\mathrm{pooled}}=\frac{\mathrm{188\; +\; 180}}{\mathrm{200\; +\; 200}}\mathrm{=\; 0.92}$

• $\mathrm{z\; =}\frac{({\hat{p}}_A-{\hat{p}}_B\mathrm{)\; -\; (0)}}{\sqrt{{\hat{p}}_{\mathrm{pooled}}\mathrm{(1\; -}{\hat{p}}_{\mathrm{pooled}}\mathrm{)(}\frac{1}{n_A}+\frac{1}{n_B})}}$
• $=\frac{\mathrm{0.94\; -\; 0.90}}{\sqrt{\mathrm{0.92(1\; -\; 0.92)(}\frac{1}{\mathrm{200}}+\frac{1}{\mathrm{200}})}}$
• $=\frac{\mathrm{0.04}}{\mathrm{0.19183}}\mathrm{=\; 2.085}$

Sketch the test statistic and critical region:

Determine the P-value

State whether you reject or fail to reject the Null hypothesis.

If you reject the null hypothesis, continue to complete the following

Calculate and display your confidence interval for the Alternative hypothesis.

State your conclusion based on your confidence interval.