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Sagot :
Let's solve this problem by following these step-by-step:
### Step 1: Calculate sample proportions
We need to determine the sample proportions from the given data:
- For the first container:
- Number of red beads sampled = 10
- Total number of beads sampled = 50
- Sample proportion ([tex]\( p1_{hat} \)[/tex]) = [tex]\( \frac{10}{50} = 0.2 \)[/tex]
- For the second container:
- Number of red beads sampled = 16
- Total number of beads sampled = 50
- Sample proportion ([tex]\( p2_{hat} \)[/tex]) = [tex]\( \frac{16}{50} = 0.32 \)[/tex]
### Step 2: Determine the combined proportion
The combined proportion ([tex]\( p_{combined} \)[/tex]) of red beads is calculated by combining the red beads and total beads from both samples:
[tex]\[ p_{combined} = \frac{10 + 16}{50 + 50} = \frac{26}{100} = 0.26 \][/tex]
### Step 3: Calculate the standard error
The standard error (SE) of the difference between the sample proportions is given by:
[tex]\[ SE = \sqrt{ p_{combined} (1 - p_{combined}) \left(\frac{1}{n1} + \frac{1}{n2}\right) } \][/tex]
where:
- [tex]\( p_{combined} = 0.26 \)[/tex]
- [tex]\( n1 = 50 \)[/tex]
- [tex]\( n2 = 50 \)[/tex]
Let's substitute the values:
[tex]\[ SE = \sqrt{ 0.26 \times (1 - 0.26) \left(\frac{1}{50} + \frac{1}{50}\right) } \][/tex]
[tex]\[ SE = \sqrt{ 0.26 \times 0.74 \left(\frac{2}{50}\right) } \][/tex]
[tex]\[ SE = \sqrt{ 0.26 \times 0.74 \times 0.04 } \][/tex]
[tex]\[ SE \approx 0.0877 \][/tex]
### Step 4: Calculate the z-score
The z-score is calculated as:
[tex]\[ z = \frac{p1_{hat} - p2_{hat}}{SE} \][/tex]
Substituting the values:
[tex]\[ z = \frac{0.2 - 0.32}{0.0877} \][/tex]
[tex]\[ z \approx -1.368 \][/tex]
### Step 5: Calculate the p-value
The p-value is the probability that the observed difference between the sample proportions is at least as extreme as the difference observed, under the null hypothesis. Given the z-score calculated, the p-value (considering a two-tailed test) can be found using a z-table:
[tex]\[ \text{p-value} = 2 \left(1 - \Phi(|z|)\right) \][/tex]
Where [tex]\(\Phi(z)\)[/tex] represents the cumulative distribution function of the normal distribution.
For [tex]\( z \approx -1.368 \)[/tex]:
[tex]\[ \text{p-value} \approx 2 ( 1 - 0.0857) = 2 \times 0.9143 \approx 0.171 \][/tex]
### Conclusion
The correct standardized test statistic and p-value for testing the hypotheses [tex]\( H_0: p1 - p2 = 0 \)[/tex] and [tex]\( H_A: p1 \neq p2 \)[/tex] are:
[tex]\[ z = \frac{0.20 - 0.32}{\sqrt{\frac{(0.26)(0.74)}{100}}}, \][/tex]
[tex]\[ \text{p-value} = 0.171 \][/tex]
Hence, the correct option is:
[tex]\[ z = \frac{0.20 - 0.32}{\sqrt{\frac{(0.20)(0.80)}{50} + \frac{(0.32)(0.68)}{50}}}, \text{ p-value } = 0.171 \][/tex]
### Step 1: Calculate sample proportions
We need to determine the sample proportions from the given data:
- For the first container:
- Number of red beads sampled = 10
- Total number of beads sampled = 50
- Sample proportion ([tex]\( p1_{hat} \)[/tex]) = [tex]\( \frac{10}{50} = 0.2 \)[/tex]
- For the second container:
- Number of red beads sampled = 16
- Total number of beads sampled = 50
- Sample proportion ([tex]\( p2_{hat} \)[/tex]) = [tex]\( \frac{16}{50} = 0.32 \)[/tex]
### Step 2: Determine the combined proportion
The combined proportion ([tex]\( p_{combined} \)[/tex]) of red beads is calculated by combining the red beads and total beads from both samples:
[tex]\[ p_{combined} = \frac{10 + 16}{50 + 50} = \frac{26}{100} = 0.26 \][/tex]
### Step 3: Calculate the standard error
The standard error (SE) of the difference between the sample proportions is given by:
[tex]\[ SE = \sqrt{ p_{combined} (1 - p_{combined}) \left(\frac{1}{n1} + \frac{1}{n2}\right) } \][/tex]
where:
- [tex]\( p_{combined} = 0.26 \)[/tex]
- [tex]\( n1 = 50 \)[/tex]
- [tex]\( n2 = 50 \)[/tex]
Let's substitute the values:
[tex]\[ SE = \sqrt{ 0.26 \times (1 - 0.26) \left(\frac{1}{50} + \frac{1}{50}\right) } \][/tex]
[tex]\[ SE = \sqrt{ 0.26 \times 0.74 \left(\frac{2}{50}\right) } \][/tex]
[tex]\[ SE = \sqrt{ 0.26 \times 0.74 \times 0.04 } \][/tex]
[tex]\[ SE \approx 0.0877 \][/tex]
### Step 4: Calculate the z-score
The z-score is calculated as:
[tex]\[ z = \frac{p1_{hat} - p2_{hat}}{SE} \][/tex]
Substituting the values:
[tex]\[ z = \frac{0.2 - 0.32}{0.0877} \][/tex]
[tex]\[ z \approx -1.368 \][/tex]
### Step 5: Calculate the p-value
The p-value is the probability that the observed difference between the sample proportions is at least as extreme as the difference observed, under the null hypothesis. Given the z-score calculated, the p-value (considering a two-tailed test) can be found using a z-table:
[tex]\[ \text{p-value} = 2 \left(1 - \Phi(|z|)\right) \][/tex]
Where [tex]\(\Phi(z)\)[/tex] represents the cumulative distribution function of the normal distribution.
For [tex]\( z \approx -1.368 \)[/tex]:
[tex]\[ \text{p-value} \approx 2 ( 1 - 0.0857) = 2 \times 0.9143 \approx 0.171 \][/tex]
### Conclusion
The correct standardized test statistic and p-value for testing the hypotheses [tex]\( H_0: p1 - p2 = 0 \)[/tex] and [tex]\( H_A: p1 \neq p2 \)[/tex] are:
[tex]\[ z = \frac{0.20 - 0.32}{\sqrt{\frac{(0.26)(0.74)}{100}}}, \][/tex]
[tex]\[ \text{p-value} = 0.171 \][/tex]
Hence, the correct option is:
[tex]\[ z = \frac{0.20 - 0.32}{\sqrt{\frac{(0.20)(0.80)}{50} + \frac{(0.32)(0.68)}{50}}}, \text{ p-value } = 0.171 \][/tex]
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