Discover a wealth of knowledge and get your questions answered at IDNLearn.com. Our platform provides accurate, detailed responses to help you navigate any topic with ease.
Sagot :
We are asked to estimate the mean height of all 9th grade students at a high school using the heights of 9 randomly selected students. Given the population standard deviation [tex]\( \sigma = 2 \)[/tex] inches, we can find the sample mean, margin of error at 90% confidence level, and the 90% confidence interval for the population mean.
Here is the detailed, step-by-step process:
### Step 1: Calculate the Sample Mean
First, we need to find the sample mean ([tex]\( \bar{x} \)[/tex]) of the 9 selected students' heights. The given heights are [tex]\( 61, 62, 71, 62, 62, 63, 63, 66, 62 \)[/tex].
[tex]\[ \bar{x} = \frac{61 + 62 + 71 + 62 + 62 + 63 + 63 + 66 + 62}{9} = \frac{512}{9} \approx 63.56 \][/tex]
### Step 2: Calculate the Margin of Error
To calculate the margin of error (ME) at a 90% confidence level, we need the Z-score for a 90% confidence level. The Z-score for a 90% confidence level (which corresponds to 95% cumulative from the left tail) is approximately 1.645.
The formula for the margin of error is:
[tex]\[ \text{Margin of Error} (ME) = Z \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
Where [tex]\( Z \)[/tex] is the Z-score (1.645 for 90% confidence), [tex]\( \sigma \)[/tex] is the population standard deviation (2 inches), and [tex]\( n \)[/tex] is the sample size (9).
[tex]\[ ME = 1.645 \times \left( \frac{2}{\sqrt{9}} \right) = 1.645 \times \frac{2}{3} \approx 1.10 \][/tex]
### Step 3: Calculate the 90% Confidence Interval
Finally, we can calculate the 90% confidence interval for the population mean using the formula:
[tex]\[ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) \][/tex]
Substituting the values we have:
[tex]\[ \text{Confidence Interval} = (63.56 - 1.10, 63.56 + 1.10) = (62.46, 64.65) \][/tex]
### Summary
- Sample Mean ([tex]\( \bar{x} \)[/tex]) = 63.56
- Margin of Error at 90% confidence level = 1.10
- 90% Confidence Interval = (62.46, 64.65)
So, the final detailed solution is:
[tex]\[ \begin{array}{l} \bar{x}=63.56 \\ \text{Margin of error at } 90\% \text{ confidence level} = 1.10 \\ 90\% \text{ confidence interval} = \left[62.46, 64.65\right] \end{array} \][/tex]
This interval means that we are 90% confident that the true mean height of all 9th grade students at the high school lies between 62.46 inches and 64.65 inches.
Here is the detailed, step-by-step process:
### Step 1: Calculate the Sample Mean
First, we need to find the sample mean ([tex]\( \bar{x} \)[/tex]) of the 9 selected students' heights. The given heights are [tex]\( 61, 62, 71, 62, 62, 63, 63, 66, 62 \)[/tex].
[tex]\[ \bar{x} = \frac{61 + 62 + 71 + 62 + 62 + 63 + 63 + 66 + 62}{9} = \frac{512}{9} \approx 63.56 \][/tex]
### Step 2: Calculate the Margin of Error
To calculate the margin of error (ME) at a 90% confidence level, we need the Z-score for a 90% confidence level. The Z-score for a 90% confidence level (which corresponds to 95% cumulative from the left tail) is approximately 1.645.
The formula for the margin of error is:
[tex]\[ \text{Margin of Error} (ME) = Z \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
Where [tex]\( Z \)[/tex] is the Z-score (1.645 for 90% confidence), [tex]\( \sigma \)[/tex] is the population standard deviation (2 inches), and [tex]\( n \)[/tex] is the sample size (9).
[tex]\[ ME = 1.645 \times \left( \frac{2}{\sqrt{9}} \right) = 1.645 \times \frac{2}{3} \approx 1.10 \][/tex]
### Step 3: Calculate the 90% Confidence Interval
Finally, we can calculate the 90% confidence interval for the population mean using the formula:
[tex]\[ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) \][/tex]
Substituting the values we have:
[tex]\[ \text{Confidence Interval} = (63.56 - 1.10, 63.56 + 1.10) = (62.46, 64.65) \][/tex]
### Summary
- Sample Mean ([tex]\( \bar{x} \)[/tex]) = 63.56
- Margin of Error at 90% confidence level = 1.10
- 90% Confidence Interval = (62.46, 64.65)
So, the final detailed solution is:
[tex]\[ \begin{array}{l} \bar{x}=63.56 \\ \text{Margin of error at } 90\% \text{ confidence level} = 1.10 \\ 90\% \text{ confidence interval} = \left[62.46, 64.65\right] \end{array} \][/tex]
This interval means that we are 90% confident that the true mean height of all 9th grade students at the high school lies between 62.46 inches and 64.65 inches.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
