Math / Data & Statistics

QuestionA certain drug is used to treat asthma. In a clinical trial of the drug, 19 of 266 treated subjects experienced headaches (based on data from the manufacturer). The accompanying calculator display shows results from a test of the claim that less than $9 \%$ of treated subjects experienced headaches. Use the normal distribution as an approximation to the binomial distribution and assume a 0.05 significance level to complete parts (a) through (e) below. $\begin{array}{l} 1 \text { 1-PropZTest } \\ \text { prop }<0.09 \\ z=-1.058386227 \\ p=0.1449396980 \\ \hat{p}=0.0714285714 \\ n=266 \end{array}$ (Round to two decimal places as needed.) c. What is the P -value? $P$ -value $=0.1449$ (Round to four decimal places as needed.) d. What is the null hypothesis, and what do you conclude about it?
Identify the null hypothesis. A. $H_{0}: p<0.09$ B. $H_{0}: p>0.09$ C. $H_{0}: p \neq 0.09$ D. $H_{0}: p=0.09$ View an example Get more help - Clear all Check answer

Studdy Solution

STEP 1

1. The sample size is $n = 266$ .
2. The number of subjects experiencing headaches is $x = 19$ .
3. The sample proportion is $\hat{p} = \frac{19}{266}$ .
4. The claim is that less than $9\%$ of treated subjects experienced headaches.
5. The significance level is $\alpha = 0.05$ .
6. The normal distribution is used as an approximation to the binomial distribution.

STEP 2

1. Identify the null and alternative hypotheses.
2. Determine the P-value from the test results.
3. Compare the P-value to the significance level.
4. Draw a conclusion about the null hypothesis.

STEP 3

Identify the null hypothesis ( $H_0$ ) and the alternative hypothesis ( $H_a$ ).
The null hypothesis ( $H_0$ ) is that the proportion of treated subjects who experience headaches is equal to $9\%$ .
$H_0: p = 0.09$
The alternative hypothesis ( $H_a$ ) is that the proportion of treated subjects who experience headaches is less than $9\%$ .
$H_a: p < 0.09$

STEP 4

The P-value is given in the calculator display as $p = 0.1449$ .

STEP 5

Compare the P-value to the significance level ( $\alpha = 0.05$ ).
Since $p = 0.1449$ is greater than $\alpha = 0.05$ , we do not reject the null hypothesis.

STEP 6

Draw a conclusion about the null hypothesis.
Since the P-value is greater than the significance level, there is not enough evidence to support the claim that less than $9\%$ of treated subjects experience headaches. Therefore, we do not reject the null hypothesis.
The null hypothesis is:
$H_0: p = 0.09$
The conclusion is that we do not reject the null hypothesis.

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