Sure! Let's walk through the process for solving this problem step-by-step.
### Step 1: Calculating the Sample Mean ([tex]\(\bar{x}\)[/tex])
The heights of the 7 randomly selected students are:
[tex]\[ 71, 75, 75, 61, 69, 65, 60 \][/tex]
To find the sample mean, we sum these values and divide by the number of students:
[tex]\[ \bar{x} = \frac{71 + 75 + 75 + 61 + 69 + 65 + 60}{7} \][/tex]
Adding these values together:
[tex]\[ 71 + 75 + 75 + 61 + 69 + 65 + 60 = 476 \][/tex]
Now, dividing by the sample size ([tex]\(n = 7\)[/tex]):
[tex]\[ \bar{x} = \frac{476}{7} \approx 68.0 \][/tex]
### Step 2: Calculating the Margin of Error at 99% Confidence Level
Given:
- The population standard deviation ([tex]\(\sigma\)[/tex]) is 3 inches.
- The sample size ([tex]\(n\)[/tex]) is 7.
- The confidence level is 99%.
To find the margin of error, we need the standard error (SE) and the z-score corresponding to the 99% confidence level.
First, let's compute the standard error (SE):
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{3}{\sqrt{7}} \][/tex]
Next, we use the z-score for a 99% confidence level, which is approximately 2.576 (this value comes from standard normal distribution tables).
The margin of error (ME) is:
[tex]\[ ME = z \cdot SE \approx 2.576 \cdot \frac{3}{\sqrt{7}} \approx 2.92 \][/tex]
### Step 3: Calculating the 99% Confidence Interval
The confidence interval is calculated by adding and subtracting the margin of error from the sample mean.
The lower bound of the confidence interval is:
[tex]\[ \text{Lower bound} = \bar{x} - ME \approx 68.0 - 2.92 \approx 65.08 \][/tex]
The upper bound of the confidence interval is:
[tex]\[ \text{Upper bound} = \bar{x} + ME \approx 68.0 + 2.92 \approx 70.92 \][/tex]
### Final Results
- Sample Mean ([tex]\(\bar{x}\)[/tex]): [tex]\(68.0\)[/tex]
- Margin of error at 99% confidence level: [tex]\(2.92\)[/tex]
- 99% confidence interval: [tex]\([65.08, 70.92]\)[/tex]
So, based on the sample data, we estimate that the mean height of all 9th grade students at the high school is 68.0 inches, with a margin of error of 2.92 inches at the 99% confidence level. This gives us a 99% confidence interval of [65.08, 70.92] inches.
