Chapter 10, Problem 5CRE

(a)

To determine

To find: out the test of the claim at the $\alpha =0.05$ level

Answer to Problem 5CRE

Yes

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=38\\ x_2=17\\ n_1=n_2=435\end{array}$

Formula used:

  $z=\frac{{\widehat{p}}_1+{\widehat{p}}_2}{\sqrt{{\widehat{p}}_p\left(1-{\widehat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

Calculation:

  $\begin{array}{c}{H}_0:p_1=p_2\\ H_1:p_1>p_2\end{array}$

The sample proportion is

  $\begin{array}{c}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{38}{435}=0.0874\end{array}$

  $\begin{array}{c}{\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{17}{435}=0.0391\end{array}$

  $\begin{array}{c}{\widehat{p}}_p=\frac{x_1+x_2}{n_1+n_2}\\ =\frac{38+17}{435+435}\\ =0.0632\end{array}$

The test-statistics is

  $\begin{array}{c}z=\frac{{\widehat{p}}_1+{\widehat{p}}_2}{\sqrt{{\widehat{p}}_p\left(1-{\widehat{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}\\ =\frac{0.0874-0.0391}{\sqrt{0.0632\left(1-0.0632\right)}\sqrt{\frac{1}{435}+\frac{1}{435}}}\\ =2.93\end{array}$

P-value is the probability of getting the value of test statistic. The P-value is

  $\begin{array}{c}P=P\left(Z>2.93\right)\\ =P\left(Z<-2.93\right)\\ =0.0017\end{array}$

When the P-value is smaller than the significance level, then reject the null hypothesis.

  $P<0.05$ therefore, here is enough proof to help the claim.

(b)

To determine

To Explain: that the double-blind experiment is important.

Explanation of Solution

A double-blind is one of the experiments in which neither the individuals nor the subjects who measure know which treatment was received, where in the single-blind experiment either the individuals who measure or the subjects know which treatment was received. Therefore, it is important for the subjects, such that it is chance to eliminate the placebo effect. This is also good for the individuals who execute who execute the measurements.

(c)

To determine

To Explain: the type I and type II error and find the serious one.

Answer to Problem 5CRE

Type-I error

Explanation of Solution

Given:

  $\begin{array}{c}{H}_0:p_1=p_2\\ H_1:p_1>p_2\end{array}$

Type I error: reject the null hypothesis, if the null hypothesis

The drug appears to be effective, where it is not effective. People would take the drug without getting better.

Type II error: fail to reject the null hypothesis, if the null hypothesis is false.

The drug appears to be not effective, if it is effective. The drug would not come on the market, the reason is that the drug appears to be not effective.

Therefore Type-I error is the most serious one, because people would take a drug without getting better.