IDNLearn.com is your reliable source for expert answers and community insights. Find the information you need quickly and easily with our comprehensive and accurate Q&A platform.
Sagot :
Let's go step-by-step to answer the questions based on the given data:
### Step 1: Identify the null and alternative hypotheses
Given that we are comparing two proportions from different populations, the two hypotheses can be stated as:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex] (the population proportions are the same)
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex] (the population proportions are not the same)
Therefore, the correct hypotheses are:
[tex]\[ H_0: p_1 = p_2 \\ H_1: p_1 \neq p_2 \][/tex]
### Step 2: Calculate the sample proportions
- The first sample consists of 20 people, with 11 having the attribute.
- The second sample consists of 2000 people, with 1447 having the attribute.
The sample proportions are:
[tex]\[ p_1 = \frac{11}{20} = 0.55 \][/tex]
[tex]\[ p_2 = \frac{1447}{2000} = 0.7235 \][/tex]
### Step 3: Calculate the combined proportion
The combined proportion [tex]\( \hat{p} \)[/tex] is calculated as:
[tex]\[ \hat{p} = \frac{11 + 1447}{20 + 2000} = \frac{1458}{2020} \approx 0.7218 \][/tex]
### Step 4: Calculate the standard error
The standard error (SE) is computed using the combined proportion:
[tex]\[ SE = \sqrt{\hat{p} \cdot (1 - \hat{p}) \cdot \left(\frac{1}{20} + \frac{1}{2000}\right)} = \sqrt{0.7218 \cdot (1 - 0.7218) \cdot \left(\frac{1}{20} + \frac{1}{2000}\right)} \approx 0.1007 \][/tex]
### Step 5: Calculate the test statistic (Z)
The Z statistic is calculated as follows:
[tex]\[ Z = \frac{p_1 - p_2}{SE} = \frac{0.55 - 0.7235}{0.1007} \approx -1.72 \][/tex]
### Step 6: Identify the critical value
For a two-tailed test at a 0.05 significance level, the critical value can be found using the Z-distribution:
[tex]\[ Z_{\text{critical}} = \pm 1.96 \][/tex]
### Step 7: Compare the test statistic to the critical value
The test statistic [tex]\( Z \approx -1.72 \)[/tex] does not exceed the critical value [tex]\( \pm 1.96 \)[/tex] in absolute terms. Therefore, we do not reject the null hypothesis.
Thus, we conclude that there is insufficient evidence to claim that the population proportions [tex]\(p_1\)[/tex] and [tex]\(p_2\)[/tex] are different at the 0.05 significance level.
### Results Summary:
- Test Statistic: [tex]\( Z \approx -1.72 \)[/tex] (rounded to two decimal places)
- Critical Values: [tex]\( \pm 1.960 \)[/tex] (rounded to three decimal places)
- Null Hypothesis: [tex]\( H_0: p_1 = p_2 \)[/tex]
- Alternative Hypothesis: [tex]\( H_1: p_1 \neq p_2 \)[/tex]
### Step 1: Identify the null and alternative hypotheses
Given that we are comparing two proportions from different populations, the two hypotheses can be stated as:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\( p_1 = p_2 \)[/tex] (the population proportions are the same)
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): [tex]\( p_1 \neq p_2 \)[/tex] (the population proportions are not the same)
Therefore, the correct hypotheses are:
[tex]\[ H_0: p_1 = p_2 \\ H_1: p_1 \neq p_2 \][/tex]
### Step 2: Calculate the sample proportions
- The first sample consists of 20 people, with 11 having the attribute.
- The second sample consists of 2000 people, with 1447 having the attribute.
The sample proportions are:
[tex]\[ p_1 = \frac{11}{20} = 0.55 \][/tex]
[tex]\[ p_2 = \frac{1447}{2000} = 0.7235 \][/tex]
### Step 3: Calculate the combined proportion
The combined proportion [tex]\( \hat{p} \)[/tex] is calculated as:
[tex]\[ \hat{p} = \frac{11 + 1447}{20 + 2000} = \frac{1458}{2020} \approx 0.7218 \][/tex]
### Step 4: Calculate the standard error
The standard error (SE) is computed using the combined proportion:
[tex]\[ SE = \sqrt{\hat{p} \cdot (1 - \hat{p}) \cdot \left(\frac{1}{20} + \frac{1}{2000}\right)} = \sqrt{0.7218 \cdot (1 - 0.7218) \cdot \left(\frac{1}{20} + \frac{1}{2000}\right)} \approx 0.1007 \][/tex]
### Step 5: Calculate the test statistic (Z)
The Z statistic is calculated as follows:
[tex]\[ Z = \frac{p_1 - p_2}{SE} = \frac{0.55 - 0.7235}{0.1007} \approx -1.72 \][/tex]
### Step 6: Identify the critical value
For a two-tailed test at a 0.05 significance level, the critical value can be found using the Z-distribution:
[tex]\[ Z_{\text{critical}} = \pm 1.96 \][/tex]
### Step 7: Compare the test statistic to the critical value
The test statistic [tex]\( Z \approx -1.72 \)[/tex] does not exceed the critical value [tex]\( \pm 1.96 \)[/tex] in absolute terms. Therefore, we do not reject the null hypothesis.
Thus, we conclude that there is insufficient evidence to claim that the population proportions [tex]\(p_1\)[/tex] and [tex]\(p_2\)[/tex] are different at the 0.05 significance level.
### Results Summary:
- Test Statistic: [tex]\( Z \approx -1.72 \)[/tex] (rounded to two decimal places)
- Critical Values: [tex]\( \pm 1.960 \)[/tex] (rounded to three decimal places)
- Null Hypothesis: [tex]\( H_0: p_1 = p_2 \)[/tex]
- Alternative Hypothesis: [tex]\( H_1: p_1 \neq p_2 \)[/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.
