$$ 1 \leftarrow \quad \begin{array}{l} ext { A certain drug is used to treat asthma. In a clinical trial of the drug, } 28 ext { of } 290 ext { treated subjects experienced } \ ext { headaches (based on data from the manufacturer). The accompanying calculator display shows results } \ ext { from a test of the claim that less than } 10 \% ext { of treated subjects experienced headaches. Use the normal } \ ext { distribution as an approximation to the binomial distribution and assume a } 0.05 ext { significance level to } \ ext { complete parts (a) through (e) below. }\end{array} $$\begin{tabular}{|c}$$ 1- $$ PropzTest \ prop $$

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$$ 1 \leftarrow \quad \begin{array}{l}	ext { A certain drug is used to treat asthma. In a clinical trial of the drug, } 28 	ext { of } 290 	ext { treated subjects experienced } \ 	ext { headaches (based on data from the manufacturer). The accompanying calculator display shows results } \ 	ext { from a test of the claim that less than } 10 \% 	ext { of treated subjects experienced headaches. Use the normal } \ 	ext { distribution as an approximation to the binomial distribution and assume a } 0.05 	ext { significance level to } \ 	ext { complete parts (a) through (e) below. }\end{array} $$\begin{tabular}{|c}$$ 1- $$ PropzTest \ prop $$ <0.1 $$ \ $$ z=-0.195740073 $$ \ $$ p=0.4224068087 $$ \ $$ \hat{p}=0.0965517241 $$ \ $$ n=290 $$\end{tabular} a. Is the test two-tailed, left-tailed, or right-tailed? Right tailed test Two-tailed test b. What is the test statistic? z= (Round to two decimal places as needed.)

UpStudy AI · Accepted Answer

To solve the problem, we need to analyze the given information and answer the questions step by step.

### Known Conditions:
- Sample size ($$ n $$) = 290
- Number of subjects experiencing headaches = 28
- Sample proportion ($$ \hat{p} $$) = $$ \frac{28}{290} \approx 0.0966 $$
- Null hypothesis ($$ H_0 $$): $$ p \geq 0.1 $$
- Alternative hypothesis ($$ H_a $$): $$ p < 0.1 $$
- Significance level = 0.05

### Part (a): Type of Test
The alternative hypothesis states that the proportion of treated subjects experiencing headaches is less than 10%. This indicates that we are testing for a decrease in proportion. Therefore, this is a **left-tailed test**.

### Part (b): Test Statistic
The test statistic for a proportion can be calculated using the formula:

$$
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
$$

Where:
- $$ \hat{p} $$ = sample proportion
- $$ p_0 $$ = hypothesized population proportion (0.1)
- $$ n $$ = sample size

Now, we will calculate the test statistic using the values provided:

1. Calculate the standard error (SE):
   $$
   SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.1(1 - 0.1)}{290}}
   $$

2. Substitute the values into the z formula:
   $$
   z = \frac{0.0966 - 0.1}{SE}
   $$

Let's perform the calculations.

### Step 1: Calculate the Standard Error
$$
SE = \sqrt{\frac{0.1 \times 0.9}{290}} = \sqrt{\frac{0.09}{290}} \approx \sqrt{0.0003103448275862069} \approx 0.0176
$$

### Step 2: Calculate the z-value
$$
z = \frac{0.0966 - 0.1}{0.0176} \approx \frac{-0.0034}{0.0176} \approx -0.1932
$$

Now, rounding to two decimal places:
$$
z \approx -0.19
$$

### Final Answers:
- **(a)** The test is a **left-tailed test**.
- **(b)** The test statistic $$ z $$ is approximately **-0.19**.
**(a)** The test is a left-tailed test. **(b)** The test statistic $$ z $$ is **-0.19**.

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