Answer :
To determine if there is convincing evidence that more than 25% of adults describe themselves as organized, we can follow these steps:
### Step 1: Set Up Hypotheses
- Null Hypothesis (H0): The proportion of adults who describe themselves as organized is 25%, [tex]\( p = 0.25 \)[/tex].
- Alternative Hypothesis (Ha): The proportion of adults who describe themselves as organized is greater than 25%, [tex]\( p > 0.25 \)[/tex].
### Step 2: Check Conditions for Inference
Before conducting a hypothesis test, we need to check if the conditions for inference are met:
1. Random Condition: The sample must be random. In this scenario, we assume the sample of 100 adults is random.
2. 10% Condition: The sample size should be less than 10% of the population. Assuming a population size of more than 1000, the sample size of 100 is less than 10% of this hypothetical population.
3. Large Counts Condition: [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex] should both be at least 10.
- [tex]\( n \times p_0 = 100 \times 0.25 = 25 \)[/tex]
- [tex]\( n \times (1 - p_0) = 100 \times 0.75 = 75 \)[/tex]
Both values are at least 10, so the large counts condition is met.
### Step 3: Perform the Test
- Sample Size (n): 100
- Number of "Successes": 42
- Sample Proportion ([tex]\( \hat{p} \)[/tex]): [tex]\( \frac{42}{100} = 0.42 \)[/tex]
- Significance Level ([tex]\( \alpha \)[/tex]): 0.01
#### Calculate the Z-Score
The standard error is calculated using the formula:
[tex]\[ \text{Standard Error} = \sqrt{\frac{p_0 \times (1 - p_0)}{n}} \][/tex]
And the Z-score is calculated as:
[tex]\[ z = \frac{\hat{p} - p_0}{\text{Standard Error}} \][/tex]
From the calculations:
- Z-Score: Approximately 3.93
#### Determine the P-Value
The P-value represents the probability of obtaining a sample proportion as extreme as 0.42, assuming the null hypothesis is true. For a Z-score of 3.93, the P-value is approximately 0.000043.
### Step 4: Make a Decision
Since the P-value (0.000043) is less than the significance level ([tex]\( \alpha = 0.01 \)[/tex]), we reject the null hypothesis.
### Conclusion
There is convincing evidence that more than 25% of adults would describe themselves as organized.
### Step 1: Set Up Hypotheses
- Null Hypothesis (H0): The proportion of adults who describe themselves as organized is 25%, [tex]\( p = 0.25 \)[/tex].
- Alternative Hypothesis (Ha): The proportion of adults who describe themselves as organized is greater than 25%, [tex]\( p > 0.25 \)[/tex].
### Step 2: Check Conditions for Inference
Before conducting a hypothesis test, we need to check if the conditions for inference are met:
1. Random Condition: The sample must be random. In this scenario, we assume the sample of 100 adults is random.
2. 10% Condition: The sample size should be less than 10% of the population. Assuming a population size of more than 1000, the sample size of 100 is less than 10% of this hypothetical population.
3. Large Counts Condition: [tex]\( n \times p_0 \)[/tex] and [tex]\( n \times (1 - p_0) \)[/tex] should both be at least 10.
- [tex]\( n \times p_0 = 100 \times 0.25 = 25 \)[/tex]
- [tex]\( n \times (1 - p_0) = 100 \times 0.75 = 75 \)[/tex]
Both values are at least 10, so the large counts condition is met.
### Step 3: Perform the Test
- Sample Size (n): 100
- Number of "Successes": 42
- Sample Proportion ([tex]\( \hat{p} \)[/tex]): [tex]\( \frac{42}{100} = 0.42 \)[/tex]
- Significance Level ([tex]\( \alpha \)[/tex]): 0.01
#### Calculate the Z-Score
The standard error is calculated using the formula:
[tex]\[ \text{Standard Error} = \sqrt{\frac{p_0 \times (1 - p_0)}{n}} \][/tex]
And the Z-score is calculated as:
[tex]\[ z = \frac{\hat{p} - p_0}{\text{Standard Error}} \][/tex]
From the calculations:
- Z-Score: Approximately 3.93
#### Determine the P-Value
The P-value represents the probability of obtaining a sample proportion as extreme as 0.42, assuming the null hypothesis is true. For a Z-score of 3.93, the P-value is approximately 0.000043.
### Step 4: Make a Decision
Since the P-value (0.000043) is less than the significance level ([tex]\( \alpha = 0.01 \)[/tex]), we reject the null hypothesis.
### Conclusion
There is convincing evidence that more than 25% of adults would describe themselves as organized.
