Find solutions to your problems with the expert advice available on IDNLearn.com. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.
Sagot :
To solve this problem, we need to calculate the 95% confidence interval for the estimated proportion (denoted [tex]\(\hat{p}\)[/tex]) of customers that come into the store because of the coffee shop.
Here is the step-by-step solution:
1. Identify the Sample Size and Sample Proportion:
- The sample size ([tex]\(n\)[/tex]) is 75.
- The sample proportion ([tex]\(\hat{p}\)[/tex]) is 0.123.
2. Determine the Z-Score for the Desired Confidence Level:
- For a 95% confidence interval, the Z-score ([tex]\(z^*\)[/tex]) is 1.960. This value is obtained from the standard normal distribution corresponding to a 95% confidence level.
3. Calculate the Standard Error:
- The standard error (SE) of the sample proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
- Plugging in the values:
[tex]\[ SE = \sqrt{\frac{0.123 \cdot (1 - 0.123)}{75}} \][/tex]
4. Calculate the Margin of Error:
- The margin of error (ME) is given by:
[tex]\[ ME = z^* \cdot SE \][/tex]
- Using the values:
[tex]\[ ME = 1.960 \cdot SE \][/tex]
5. Determine the Confidence Interval:
- The lower bound of the confidence interval is:
[tex]\[ \text{Lower bound} = \hat{p} - ME \][/tex]
- The upper bound of the confidence interval is:
[tex]\[ \text{Upper bound} = \hat{p} + ME \][/tex]
6. Calculate the Numerical Values:
- After calculating, we find that the standard error is approximately [tex]\(0.038\)[/tex].
- The margin of error then is approximately [tex]\(0.074\)[/tex].
- Therefore, the confidence interval is:
[tex]\[ \text{Lower bound} = 0.123 - 0.074 = 0.049 \][/tex]
[tex]\[ \text{Upper bound} = 0.123 + 0.074 = 0.197 \][/tex]
Thus, the manager can say with 95% confidence that the true population proportion of customers that come into the store because of the coffee shop is in the interval [tex]\((0.049, 0.197)\)[/tex].
### Summary
The 95% confidence interval for the proportion of customers that come into the store because of the coffee shop is:
- Upper bound: 0.197
- Lower bound: 0.049
Therefore, you can state that with 95% confidence, the true population proportion is in the interval (0.049, 0.197).
Here is the step-by-step solution:
1. Identify the Sample Size and Sample Proportion:
- The sample size ([tex]\(n\)[/tex]) is 75.
- The sample proportion ([tex]\(\hat{p}\)[/tex]) is 0.123.
2. Determine the Z-Score for the Desired Confidence Level:
- For a 95% confidence interval, the Z-score ([tex]\(z^*\)[/tex]) is 1.960. This value is obtained from the standard normal distribution corresponding to a 95% confidence level.
3. Calculate the Standard Error:
- The standard error (SE) of the sample proportion is calculated using the formula:
[tex]\[ SE = \sqrt{\frac{\hat{p} \cdot (1 - \hat{p})}{n}} \][/tex]
- Plugging in the values:
[tex]\[ SE = \sqrt{\frac{0.123 \cdot (1 - 0.123)}{75}} \][/tex]
4. Calculate the Margin of Error:
- The margin of error (ME) is given by:
[tex]\[ ME = z^* \cdot SE \][/tex]
- Using the values:
[tex]\[ ME = 1.960 \cdot SE \][/tex]
5. Determine the Confidence Interval:
- The lower bound of the confidence interval is:
[tex]\[ \text{Lower bound} = \hat{p} - ME \][/tex]
- The upper bound of the confidence interval is:
[tex]\[ \text{Upper bound} = \hat{p} + ME \][/tex]
6. Calculate the Numerical Values:
- After calculating, we find that the standard error is approximately [tex]\(0.038\)[/tex].
- The margin of error then is approximately [tex]\(0.074\)[/tex].
- Therefore, the confidence interval is:
[tex]\[ \text{Lower bound} = 0.123 - 0.074 = 0.049 \][/tex]
[tex]\[ \text{Upper bound} = 0.123 + 0.074 = 0.197 \][/tex]
Thus, the manager can say with 95% confidence that the true population proportion of customers that come into the store because of the coffee shop is in the interval [tex]\((0.049, 0.197)\)[/tex].
### Summary
The 95% confidence interval for the proportion of customers that come into the store because of the coffee shop is:
- Upper bound: 0.197
- Lower bound: 0.049
Therefore, you can state that with 95% confidence, the true population proportion is in the interval (0.049, 0.197).
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.
