IDNLearn.com is your go-to platform for finding reliable answers quickly. Our experts provide timely and accurate responses to help you navigate any topic or issue with confidence.
Sagot :
Sure, let's go through this step-by-step to determine the mean height, margin of error, and the confidence interval at a 90% confidence level.
### 1. Calculate the sample mean ( [tex]\(\bar{x}\)[/tex] ):
The sample data given is:
[tex]\[71, 68, 73, 60, 62, 74, 68, 68, 61, 60\][/tex]
To find the sample mean, you add up all the heights and divide by the number of students (sample size):
[tex]\[ \bar{x} = \frac{71 + 68 + 73 + 60 + 62 + 74 + 68 + 68 + 61 + 60}{10} = \frac{665}{10} = 66.5 \][/tex]
### 2. Calculate the margin of error at a 90% confidence level:
The margin of error (ME) can be calculated using the formula:
[tex]\[ ME = Z_{\alpha/2} \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
where:
- [tex]\( Z_{\alpha/2} \)[/tex] is the z-score corresponding to the desired confidence level.
- [tex]\( \sigma \)[/tex] is the population standard deviation.
- [tex]\( n \)[/tex] is the sample size.
Given:
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 2 inches
- Sample size ([tex]\(n\)[/tex]) = 10
- Confidence level = 90%
For a 90% confidence level, the z-score ([tex]\( Z_{\alpha/2} \)[/tex]) is approximately 1.645.
Now, plug these values into the margin of error formula:
[tex]\[ ME = 1.645 \times \left( \frac{2}{\sqrt{10}} \right) = 1.645 \times 0.6325 = 1.0402967757511148 \][/tex]
### 3. Calculate the 90% confidence interval:
The confidence interval is calculated as:
[tex]\[ \left( \bar{x} - ME, \bar{x} + ME \right) \][/tex]
Using the sample mean ([tex]\(\bar{x}\)[/tex]) and the margin of error (ME) we calculated:
[tex]\[ \text{Confidence interval} = (66.5 - 1.0402967757511148, 66.5 + 1.0402967757511148) = (65.45970322424888, 67.54029677575112) \][/tex]
### Summary
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 66.5
- Margin of error at 90% confidence level = 1.0402967757511148
- 90% confidence interval = [tex]\([65.45970322424888, 67.54029677575112]\)[/tex]
Thus, the detailed calculations provide:
[tex]\[ \bar{x} = 66.5 \][/tex]
Margin of error at [tex]\(90\% \)[/tex] confidence level [tex]\(= 1.0402967757511148 \)[/tex]
[tex]\[ 90 \% \text{ confidence interval } = [65.45970322424888, 67.54029677575112] \][/tex]
### 1. Calculate the sample mean ( [tex]\(\bar{x}\)[/tex] ):
The sample data given is:
[tex]\[71, 68, 73, 60, 62, 74, 68, 68, 61, 60\][/tex]
To find the sample mean, you add up all the heights and divide by the number of students (sample size):
[tex]\[ \bar{x} = \frac{71 + 68 + 73 + 60 + 62 + 74 + 68 + 68 + 61 + 60}{10} = \frac{665}{10} = 66.5 \][/tex]
### 2. Calculate the margin of error at a 90% confidence level:
The margin of error (ME) can be calculated using the formula:
[tex]\[ ME = Z_{\alpha/2} \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
where:
- [tex]\( Z_{\alpha/2} \)[/tex] is the z-score corresponding to the desired confidence level.
- [tex]\( \sigma \)[/tex] is the population standard deviation.
- [tex]\( n \)[/tex] is the sample size.
Given:
- Population standard deviation ([tex]\(\sigma\)[/tex]) = 2 inches
- Sample size ([tex]\(n\)[/tex]) = 10
- Confidence level = 90%
For a 90% confidence level, the z-score ([tex]\( Z_{\alpha/2} \)[/tex]) is approximately 1.645.
Now, plug these values into the margin of error formula:
[tex]\[ ME = 1.645 \times \left( \frac{2}{\sqrt{10}} \right) = 1.645 \times 0.6325 = 1.0402967757511148 \][/tex]
### 3. Calculate the 90% confidence interval:
The confidence interval is calculated as:
[tex]\[ \left( \bar{x} - ME, \bar{x} + ME \right) \][/tex]
Using the sample mean ([tex]\(\bar{x}\)[/tex]) and the margin of error (ME) we calculated:
[tex]\[ \text{Confidence interval} = (66.5 - 1.0402967757511148, 66.5 + 1.0402967757511148) = (65.45970322424888, 67.54029677575112) \][/tex]
### Summary
- Sample mean ([tex]\( \bar{x} \)[/tex]) = 66.5
- Margin of error at 90% confidence level = 1.0402967757511148
- 90% confidence interval = [tex]\([65.45970322424888, 67.54029677575112]\)[/tex]
Thus, the detailed calculations provide:
[tex]\[ \bar{x} = 66.5 \][/tex]
Margin of error at [tex]\(90\% \)[/tex] confidence level [tex]\(= 1.0402967757511148 \)[/tex]
[tex]\[ 90 \% \text{ confidence interval } = [65.45970322424888, 67.54029677575112] \][/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.
