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The 80% confidence interval for the true population mean \( \mu \) is (32.1, 35.9).
Solución
To construct a confidence interval for the true population mean \( \mu \) at an \( 80\% \) confidence level, we can use the formula for the confidence interval:
\[
\bar{x} \pm z \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \( \bar{x} \) is the sample mean,
- \( z \) is the z-score corresponding to the desired confidence level,
- \( s \) is the sample standard deviation,
- \( n \) is the sample size.
### Step 1: Identify the known values
- Sample size \( n = 29 \)
- Sample mean \( \bar{x} = 34 \)
- Sample standard deviation \( s = 8 \)
### Step 2: Find the z-score for \( 80\% \) confidence level
For an \( 80\% \) confidence level, the z-score can be found using the standard normal distribution. The critical value \( z \) for \( 80\% \) confidence level (which leaves \( 10\% \) in each tail) is approximately \( 1.28 \).
### Step 3: Calculate the standard error (SE)
The standard error (SE) is calculated as follows:
\[
SE = \frac{s}{\sqrt{n}} = \frac{8}{\sqrt{29}}
\]
### Step 4: Calculate the confidence interval
Now we can calculate the confidence interval using the values we have:
\[
\text{Margin of Error} = z \cdot SE
\]
Finally, we will compute the confidence interval:
\[
\text{Confidence Interval} = \bar{x} \pm \text{Margin of Error}
\]
Let's perform the calculations step by step.
1. Calculate the standard error (SE).
2. Calculate the margin of error.
3. Construct the confidence interval.
I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
\(\frac{8}{\sqrt{29}}\)
- step1: Multiply by the Conjugate:
\(\frac{8\sqrt{29}}{\sqrt{29}\times \sqrt{29}}\)
- step2: Multiply the numbers:
\(\frac{8\sqrt{29}}{29}\)
Calculate or simplify the expression \( 1.28*(8/\sqrt(29)) \).
Calculate the value by following steps:
- step0: Calculate:
\(1.28\left(\frac{8}{\sqrt{29}}\right)\)
- step1: Remove the parentheses:
\(1.28\times \frac{8}{\sqrt{29}}\)
- step2: Convert the expressions:
\(\frac{32}{25}\times \frac{8}{\sqrt{29}}\)
- step3: Multiply the numbers:
\(\frac{256}{25\sqrt{29}}\)
- step4: Multiply by the Conjugate:
\(\frac{256\sqrt{29}}{25\sqrt{29}\times \sqrt{29}}\)
- step5: Multiply the numbers:
\(\frac{256\sqrt{29}}{725}\)
### Step 3: Calculate the Standard Error (SE)
The standard error (SE) is calculated as follows:
\[
SE = \frac{s}{\sqrt{n}} \approx 1.5
\]
### Step 4: Calculate the Margin of Error
Now, we calculate the margin of error using the z-score:
\[
\text{Margin of Error} = z \cdot SE \approx 1.28 \cdot 1.5 \approx 1.9
\]
### Step 5: Construct the Confidence Interval
Now we can construct the confidence interval:
\[
\text{Confidence Interval} = \bar{x} \pm \text{Margin of Error} = 34 \pm 1.9
\]
Calculating the lower and upper bounds:
- Lower bound: \( 34 - 1.9 = 32.1 \)
- Upper bound: \( 34 + 1.9 = 35.9 \)
### Final Answer
The \( 80\% \) confidence interval to estimate the true population mean \( \mu \) is:
\[
(32.1, 35.9)
\]
Thus, the confidence interval is \( (32.1, 35.9) \).
Revisado y aprobado por el equipo de tutoría de UpStudy
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