Suppose the mean height in inches of all 9th grade students at one high school is estimated. The population standard deviation is 5 inches. The heights of 9 randomly selected students are 70, 69, 63, 73, 67, 65, 70, 63 and 73. Ex: 12.34 %3D Margin of error at 95% confidence level = Ex: 1.23 95% confidence interval = [ Ex: 12.34 Ex: 12.34 ] [smaller value, larger value]
Suppose the mean height in inches of all 9th grade students at one high school is estimated. The population standard deviation is 5 inches. The heights of 9 randomly selected students are 70, 69, 63, 73, 67, 65, 70, 63 and 73. Ex: 12.34 %3D Margin of error at 95% confidence level = Ex: 1.23 95% confidence interval = [ Ex: 12.34 Ex: 12.34 ] [smaller value, larger value]
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![### Estimating the Mean Height of 9th Grade Students
To estimate the mean height of all 9th grade students at one high school, consider the following information and calculations.
**Given Data:**
- Population standard deviation (σ): 5 inches
- Heights of 9 randomly selected students (in inches): 70, 69, 63, 73, 67, 65, 70, 63, 73
**Sample Mean (\(\bar{x}\)):**
The sample mean height is calculated by taking the average of the heights of the 9 students.
\[ \bar{x} = \frac{(70 + 69 + 63 + 73 + 67 + 65 + 70 + 63 + 73)}{9} \approx 68.11 \text{ inches} \]
**Margin of Error at 95% Confidence Level:**
The margin of error (E) can be calculated using the formula for the confidence interval for a mean:
\[ E = z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( z \) is the z-value corresponding to the 95% confidence level (approximately 1.96).
- \( \sigma \) is the population standard deviation (5 inches).
- \( n \) is the sample size (9).
\[ E = 1.96 \left(\frac{5}{\sqrt{9}}\right) = 1.96 \times \frac{5}{3} \approx 3.27 \text{ inches} \]
**95% Confidence Interval:**
The 95% confidence interval for the population mean is calculated as:
\[ CI = \left[ \bar{x} - E, \bar{x} + E \right] \]
Substituting the values:
\[ CI \approx \left[ 68.11 - 3.27, 68.11 + 3.27 \right] \]
\[ CI \approx \left[ 64.84, 71.38 \right] \]
**Summary:**
- Sample Mean (\(\bar{x}\)): ~68.11 inches
- Margin of Error (E) at 95% confidence level: ~3.27 inches
- 95% Confidence Interval: [64.84 inches, 71.38 inches]
This interval suggests that we are 95% confident that](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe161b52d-48e0-4a6a-8396-c379dba0515b%2F4542b1c1-2ca2-474c-b44f-c36b8a84a51b%2Fznjbsbo_processed.png&w=3840&q=75)
Transcribed Image Text:### Estimating the Mean Height of 9th Grade Students
To estimate the mean height of all 9th grade students at one high school, consider the following information and calculations.
**Given Data:**
- Population standard deviation (σ): 5 inches
- Heights of 9 randomly selected students (in inches): 70, 69, 63, 73, 67, 65, 70, 63, 73
**Sample Mean (\(\bar{x}\)):**
The sample mean height is calculated by taking the average of the heights of the 9 students.
\[ \bar{x} = \frac{(70 + 69 + 63 + 73 + 67 + 65 + 70 + 63 + 73)}{9} \approx 68.11 \text{ inches} \]
**Margin of Error at 95% Confidence Level:**
The margin of error (E) can be calculated using the formula for the confidence interval for a mean:
\[ E = z \left(\frac{\sigma}{\sqrt{n}}\right) \]
Where:
- \( z \) is the z-value corresponding to the 95% confidence level (approximately 1.96).
- \( \sigma \) is the population standard deviation (5 inches).
- \( n \) is the sample size (9).
\[ E = 1.96 \left(\frac{5}{\sqrt{9}}\right) = 1.96 \times \frac{5}{3} \approx 3.27 \text{ inches} \]
**95% Confidence Interval:**
The 95% confidence interval for the population mean is calculated as:
\[ CI = \left[ \bar{x} - E, \bar{x} + E \right] \]
Substituting the values:
\[ CI \approx \left[ 68.11 - 3.27, 68.11 + 3.27 \right] \]
\[ CI \approx \left[ 64.84, 71.38 \right] \]
**Summary:**
- Sample Mean (\(\bar{x}\)): ~68.11 inches
- Margin of Error (E) at 95% confidence level: ~3.27 inches
- 95% Confidence Interval: [64.84 inches, 71.38 inches]
This interval suggests that we are 95% confident that
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