You wish to test the following claim (H) at a significance level of a = 0.10. H.:p = 0.11 H:p< 0.11 You obtain a sample of size n 169 in which there are 8 successful observations. Determine the test statistic formula for this test. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... O less than (or equal to) a greater than a This test statistic leads to a decision to... O reject the null O accept the null

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# Hypothesis Testing Example

### Problem Statement
You wish to test the following claim (\( H_a \)) at a significance level of \( \alpha = 0.10 \):

- \( H_0 \): \( p = 0.11 \)
- \( H_a \): \( p < 0.11 \)

You obtain a sample of size \( n = 169 \) in which there are 8 successful observations.

### Steps for Hypothesis Testing

#### 1. Determine the test statistic formula for this test.

   The test statistic for a proportion is usually given by:
   \[
   z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
   \]
   where:
   - \(\hat{p}\) is the sample proportion
   - \(p_0\) is the hypothesized population proportion
   - \(n\) is the sample size

#### 2. Calculate the test statistic for this sample.
   (Report answer accurate to three decimal places.)

   \[
   \text{test statistic} = \_\_\_\_\_
   \]

#### 3. Determine the p-value for this sample.
   (Report answer accurate to four decimal places.)

   \[
   \text{p-value} = \_\_\_\_\_
   \]

### Decision Making

The p-value is…
- \( \circ \) less than (or equal to) \( \alpha \)
- \( \circ \) greater than \( \alpha \)

This test statistic leads to a decision to…
- \( \circ \) reject the null
- \( \circ \) accept the null
- \( \circ \) fail to reject the null

### Conclusion
Based on the p-value, you'll make a decision to either reject the null hypothesis \( H_0 \) or fail to reject it, thereby supporting the alternative hypothesis \( H_a \).
Transcribed Image Text:# Hypothesis Testing Example ### Problem Statement You wish to test the following claim (\( H_a \)) at a significance level of \( \alpha = 0.10 \): - \( H_0 \): \( p = 0.11 \) - \( H_a \): \( p < 0.11 \) You obtain a sample of size \( n = 169 \) in which there are 8 successful observations. ### Steps for Hypothesis Testing #### 1. Determine the test statistic formula for this test. The test statistic for a proportion is usually given by: \[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \] where: - \(\hat{p}\) is the sample proportion - \(p_0\) is the hypothesized population proportion - \(n\) is the sample size #### 2. Calculate the test statistic for this sample. (Report answer accurate to three decimal places.) \[ \text{test statistic} = \_\_\_\_\_ \] #### 3. Determine the p-value for this sample. (Report answer accurate to four decimal places.) \[ \text{p-value} = \_\_\_\_\_ \] ### Decision Making The p-value is… - \( \circ \) less than (or equal to) \( \alpha \) - \( \circ \) greater than \( \alpha \) This test statistic leads to a decision to… - \( \circ \) reject the null - \( \circ \) accept the null - \( \circ \) fail to reject the null ### Conclusion Based on the p-value, you'll make a decision to either reject the null hypothesis \( H_0 \) or fail to reject it, thereby supporting the alternative hypothesis \( H_a \).
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