MATLAB: An Introduction with Applications
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ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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![**Confidence Interval Construction Using the t-Distribution**
**Problem Statement:**
Construct the indicated confidence interval for the population mean (μ) using the t-distribution. Assume the population is normally distributed.
- Confidence level (c) = 0.95
- Sample mean (\( \bar{x} \)) = 12.6
- Sample standard deviation (s) = 3.0
- Sample size (n) = 10
**Confidence Interval:**
\[ (\_, \_) \]
**Instructions:**
Round to one decimal place as needed.
**Understanding Confidence Intervals:**
A confidence interval provides a range of values that is likely to contain the population mean (μ) with a certain level of confidence. When the population standard deviation is unknown and the sample size is small (typically n < 30), the t-distribution is more appropriate than the normal distribution for constructing the interval. Here, we use a 95% confidence level, which means we are 95% confident that the interval calculated from the sample data contains the true population mean.
To calculate the confidence interval, you would generally follow these steps:
1. Determine the t-score for your confidence level and degrees of freedom (df = n - 1).
2. Calculate the standard error (SE) of the mean: SE = s / √n.
3. Calculate the margin of error (ME): ME = t-score * SE.
4. Determine the confidence interval: ( \( \bar{x} \) - ME, \( \bar{x} \) + ME ).
**Note:** Provide actual values in place of placeholders after computation.](https://content.bartleby.com/qna-images/question/09fc76f0-690c-4548-a807-d9841947f510/a4be107c-f5ea-49a9-900d-0832e96a0595/v8ummil_processed.png)
Transcribed Image Text:**Confidence Interval Construction Using the t-Distribution**
**Problem Statement:**
Construct the indicated confidence interval for the population mean (μ) using the t-distribution. Assume the population is normally distributed.
- Confidence level (c) = 0.95
- Sample mean (\( \bar{x} \)) = 12.6
- Sample standard deviation (s) = 3.0
- Sample size (n) = 10
**Confidence Interval:**
\[ (\_, \_) \]
**Instructions:**
Round to one decimal place as needed.
**Understanding Confidence Intervals:**
A confidence interval provides a range of values that is likely to contain the population mean (μ) with a certain level of confidence. When the population standard deviation is unknown and the sample size is small (typically n < 30), the t-distribution is more appropriate than the normal distribution for constructing the interval. Here, we use a 95% confidence level, which means we are 95% confident that the interval calculated from the sample data contains the true population mean.
To calculate the confidence interval, you would generally follow these steps:
1. Determine the t-score for your confidence level and degrees of freedom (df = n - 1).
2. Calculate the standard error (SE) of the mean: SE = s / √n.
3. Calculate the margin of error (ME): ME = t-score * SE.
4. Determine the confidence interval: ( \( \bar{x} \) - ME, \( \bar{x} \) + ME ).
**Note:** Provide actual values in place of placeholders after computation.
Expert Solution
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Step 1
Since population standard deviation is unknown, Use t-distribution to find t-critical value.
Find the t-critical value:
Refer t-distribution table and look for t value corresponding to degrees of freedom=9 & confidence level of 95%.
Therefore,
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