The uploaded image contains a statistics problem about hypothesis testing. Let’s break it down step by step:

Problem:

A drug testing company is testing for marijuana usage. Among 292 tested subjects, results from 27 were wrong (false positives or false negatives). The claim is that less than 10% of test results are wrong, and the test is conducted at a 0.01 significance level.

1. Null Hypothesis ($H_0$) and Alternative Hypothesis ($H_1$):

   • From the question, the null hypothesis ($H_0$) assumes the proportion of wrong results ($p$) is equal to 0.1 (10%).
   • The alternative hypothesis ($H_1$) claims that the proportion of wrong results ($p$) is less than 0.1.

   Thus, the correct option is: $H_0: p = 0.1 \quad \text{and} \quad H_1: p < 0.1$

2. Identify the Test Statistic: To compute the test statistic, we use the formula for a z-test for proportions: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0 (1 - p_0)}{n}}}$ Where:

   • $\hat{p} = \frac{x}{n} = \frac{27}{292}$, the sample proportion of wrong results.
   • $p_0 = 0.1$, the proportion under the null hypothesis.
   • $n = 292$, the total number of subjects.

Step 1: Calculate $\hat{p}$:

$\hat{p} = \frac{27}{292} \approx 0.09247$

Step 2: Compute the Standard Error:

$\text{Standard Error} = \sqrt{\frac{p_0 (1 - p_0)}{n}} = \sqrt{\frac{0.1 (1 - 0.1)}{292}} = \sqrt{\frac{0.09}{292}} \approx 0.01751$

Step 3: Compute the Test Statistic ($z$):

$z = \frac{\hat{p} - p_0}{\text{Standard Error}} = \frac{0.09247 - 0.1}{0.01751} \approx \frac{-0.00753}{0.01751} \approx -0.43$

Final Answer:

The test statistic is approximately: $z \approx -0.43$

Would you like a detailed explanation of the interpretation of this result, or do you have specific questions?

5 Follow-Up Questions:

1. What does the calculated $z$-value imply in terms of hypothesis testing?
2. How do we determine the p-value associated with this $z$-value?
3. What is the conclusion of the test at the 0.01 significance level?
4. How would the hypotheses change if the claim was "more than 10% of test results are wrong"?
5. Why do we use a z-test for this type of proportion problem instead of a t-test?

Tip:

When computing standard error in hypothesis tests, always ensure that $n \cdot p_0$ and $n \cdot (1 - p_0)$ are greater than 5 for the approximation to the normal distribution to be valid.