Question 1162307: A mechanical engineer is analyzing tensile strength of steel ( API 5L X65). A sample of 7 specimens showed a sample mean of 530 MPa. The standard deviation is known to be LaTeX: MPaĻ = 5 . A 95% upper confidence interval for the true mean tensile strength is:
Answer by CPhill(2264)   (Show Source):
You can put this solution on YOUR website! To find the **95% upper confidence interval** (a one-sided confidence interval) for the true mean tensile strength, we follow these steps:
### 1. Identify the given parameters
* **Sample Mean ($\bar{x}$):** 530 MPa
* **Known Population Standard Deviation ($\sigma$):** 5 MPa
* **Sample Size ($n$):** 7
* **Confidence Level:** 95% (one-sided)
### 2. Determine the Critical Value ($z_{\alpha}$)
Because the population standard deviation ($\sigma$) is known, we use the standard normal distribution ($z$-distribution). For a **one-sided** 95% upper confidence interval, the entire 5% error ($\alpha = 0.05$) is placed in the upper tail.
* The $z$-score that leaves 5% in the upper tail (or corresponds to a cumulative area of 0.95) is **$1.645$**.
### 3. Calculate the Standard Error ($SE$)
The standard error of the mean is calculated as:
$$SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{7}} \approx \frac{5}{2.6458} \approx 1.8898 \text{ MPa}$$
### 4. Calculate the Upper Bound
The formula for the 95% upper confidence interval is:
$$\text{Upper Limit} = \bar{x} + z_{\alpha} \cdot \left( \frac{\sigma}{\sqrt{n}} \right)$$
$$\text{Upper Limit} = 530 + (1.645 \cdot 1.8898)$$
$$\text{Upper Limit} = 530 + 3.1087$$
$$\text{Upper Limit} \approx 533.11 \text{ MPa}$$
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### Final Result
The 95% upper confidence interval for the true mean tensile strength is:
$$(-\infty, 533.11 \text{ MPa}]$$
**Answer:** The true mean tensile strength is less than or equal to **533.11 MPa** with 95% confidence.
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