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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]
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