QUESTION 12 In a sample of 100 people who have had strokes, the average cholesterol level was 250 with a standard deviation of 40. In order to test the hypothesis (at the $$ 5 \% $$ level of significance) that the average cholesterol level of people who have had strokes was at least 240 , what is the null and alternative hypotheses? $$ H_{0}: \mu=240, H_{A}: \mu eq 240 $$ $$ H_{0}: \mu \leq 240, H_{A}: \mu240 $$ $$ H_{0}: \mu \geq 240, H_{A}: \mu250 $$ $$ H_{0}: \bar{X} \geq 250, H_{A}: \bar{X}

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QUESTION 12 In a sample of 100 people who have had strokes, the average cholesterol level was 250 with a standard deviation of 40. In order to test the hypothesis (at the $$ 5 \% $$ level of significance) that the average cholesterol level of people who have had strokes was at least 240 , what is the null and alternative hypotheses? $$ H_{0}: \mu=240, H_{A}: \mu eq 240 $$ $$ H_{0}: \mu \leq 240, H_{A}: \mu>240 $$ $$ H_{0}: \mu \geq 240, H_{A}: \mu<240 $$ $$ H_{0}: p \geq 240, H_{A}: p<240 $$ $$ H_{0}: p \leq 240, H_{A}: p>240 $$ $$ H_{0}: p=240, H_{A}: p eq 240 $$ $$ H_{0}: \bar{p}=240, H_{A}: \bar{p} eq 240 $$ $$ H_{0}: \bar{p} \geq 240, H_{A}: \bar{p}<240 $$ $$ H_{0}: \bar{p} \leq 240, H_{A}: \bar{p}>240 $$ $$ H_{0}: \bar{X} \leq 250, H_{A}: \bar{X}>250 $$ $$ H_{0}: \bar{X} \geq 250, H_{A}: \bar{X}<250 $$ $$ H_{0}: \bar{X}=250, H_{A}: \bar{X} eq 250 $$

UpStudy AI · Accepted Answer

To test the hypothesis that the **average cholesterol level** of people who have had strokes is **at least 240**, we need to establish both the null and alternative hypotheses.

### **Definitions:**
- **Null Hypothesis ($$ H_0 $$)**: This is the statement we assume to be true until proven otherwise. It often represents a status quo or a statement of no effect.
- **Alternative Hypothesis ($$ H_A $$)**: This is what you aim to support. It represents a new effect or difference.

### **Given:**
- **Sample Size ($$ n $$)**: 100
- **Sample Mean ($$ \bar{X} $$)**: 250
- **Sample Standard Deviation ($$ s $$)**: 40
- **Hypothesized Mean ($$ \mu_0 $$)**: 240
- **Significance Level ($$ \alpha $$)**: 5%

### **Formulating the Hypotheses:**
- **Null Hypothesis ($$ H_0 $$)**: The average cholesterol level is **at least 240**. Mathematically, this is represented as:
  $$
  H_0: \mu \geq 240
  $$
  
- **Alternative Hypothesis ($$ H_A $$)**: The average cholesterol level is **less than 240**. Mathematically, this is:
  $$
  H_A: \mu < 240
  $$

This setup indicates a **one-tailed test** because we are only interested in whether the mean is **less than** 240, not just different from 240.

### **Available Options:**
Among the provided options, the correct representation of the null and alternative hypotheses is:

$$
H_0: \mu \geq 240, \quad H_A: \mu < 240
$$

This corresponds to the **third option** in your list.

### **Conclusion:**
The appropriate null and alternative hypotheses for this scenario are:

**$$ H_{0}: \mu \geq 240, H_{A}: \mu<240 $$**

**Answer:**  
**$$ H_{0}: \mu \geq 240, H_{A}: \mu<240 $$**
The null hypothesis is $$ H_{0}: \mu \geq 240 $$ and the alternative hypothesis is $$ H_{A}: \mu < 240 $$.

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