Chapter 10.1, Problem 30E

(a)

To determine

To state appropriate hypotheses for performing a significance test and also define the parameters.

Answer to Problem 30E

The hypotheses is:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Explanation of Solution

It is given that the researcher wants to study about the effect of Botox on the pain relief treatment for the patients.

So, the given claim is that: the proportion is greater for Botox.

Now, we have to find out the appropriate hypotheses for performing a significance test.

Thus, the claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportions are equal. If the null hypothesis is the claim then the alternative hypothesis states the opposite of the null hypothesis.

Therefore, the appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Where we have,

  $p_1=$ the proportion of patients who received the Botox treatment and experience pain relief.

  $p_2=$ the proportion of patients who received the Saline treatment and experience pain relief.

(b)

To determine

To explain why you should not use the method of this section to calculate the P -value.

Explanation of Solution

Given:

It is given that the researcher wants to study about the effect of Botox on the pain relief treatment for the patients.

  $\begin{array}{l}{x}_1=10\\ x_2=3\\ n_1=15\\ n_2=16\end{array}$

Conditions to be satisfied:

There are three conditions to be satisfied:

Random: It is satisfied because the patients are independent random samples.

Independent: It is satisfied because the $16$ patients who suffered from low-back pain are less than $10\%$ of all patients who suffered from low-back pain.

Normal: It is not satisfied because there are $3$ successes in the second sample, which is not at least ten.

Thus, all the conditions are not satisfied since the large costs is not satisfied and it is not appropriate to use hypothesis test for testing a claim. Thus, you should not use the method of this section to calculate the P -value.

(c)

To determine

To find out what is the estimated P -value.

Answer to Problem 30E

The estimated P values is $0.02$ .

Explanation of Solution

Given:

It is given that the researcher wants to study about the effect of Botox on the pain relief treatment for the patients.

  $\begin{array}{l}{x}_1=10\\ x_2=3\\ n_1=15\\ n_2=16\end{array}$

Hypotheses:

So, the given claim is that: the proportion is greater for Botox.

Now, we have to find out the appropriate hypotheses for performing a significance test.

Thus, the claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportions are equal. If the null hypothesis is the claim then the alternative hypothesis states the opposite of the null hypothesis.

Therefore, the appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Where we have,

  $p_1=$ the proportion of patients who received the Botox treatment and experience pain relief.

  $p_2=$ the proportion of patients who received the Saline treatment and experience pain relief.

Calculation:

The sample proportion is the number of successes divided by the sample size. Then, we have,

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{10}{15}\\ =0.6667\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{3}{16}\\ =0.1875\end{array}$

Now, we will calculate the difference of the sample proportions:

  $\begin{array}{l}{\widehat{p}}_1-{\widehat{p}}_2=0.6667-0.1875\\ =0.4792\end{array}$

The P-value is the probability of obtaining the value of the test statistics or a value more extreme assuming that the null hypothesis is true. Thus, we have,

There are two dots at $0.4792$ and no dots to the right of $0.4792$ , which implies that two of the $100$ dots in the dot plot corresponds to the difference of sample proportions of at least $0.4792$ .

  $\begin{array}{l}P\text{-value}=\frac{2}{100}\\ =0.02\end{array}$

Thus, the estimated P values is $0.02$ .

(d)

To determine

To explain what conclusion what you draw in this context.

Explanation of Solution

From part (c) we have that,

The appropriate hypotheses for this is:

  $\begin{array}{l}{H}_0:p_1=p_2\\ H_a:p_1>p_2\end{array}$

Where we have,

  $p_1=$ the proportion of patients who received the Botox treatment and experience pain relief.

  $p_2=$ the proportion of patients who received the Saline treatment and experience pain relief.

Calculation:

The sample proportion is the number of successes divided by the sample size. Then, we have,

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{10}{15}\\ =0.6667\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{3}{16}\\ =0.1875\end{array}$

Now, we will calculate the difference of the sample proportions:

  $\begin{array}{l}{\widehat{p}}_1-{\widehat{p}}_2=0.6667-0.1875\\ =0.4792\end{array}$

The P-value is the probability of obtaining the value of the test statistics or a value more extreme assuming that the null hypothesis is true. Thus, we have,

There are two dots at $0.4792$ and no dots to the right of $0.4792$ , which implies that two of the $100$ dots in the dot plot corresponds to the difference of sample proportions of at least $0.4792$ .

  $\begin{array}{l}P\text{-value}=\frac{2}{100}\\ =0.02\end{array}$

Thus, if the P-value is smaller than the significance level, then we will reject the null hypothesis, thus, we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is convincing evidence that the proportion of patients who uses Botox treatment increases the pain relief among the patients who suffered from chronic low-back pain.