SOLUTION: The mean length of a small counterbalance bar is 43 millimeters. The production supervisor is concerned that the adjustments of the machine producing the bar have changed. He asks

Question 1162408: The mean length of a small counterbalance bar is 43 millimeters. The production supervisor
is concerned that the adjustments of the machine producing the bar have changed. He asks
the engineering to investigate. Engineering selects a random sample of 30 bars and measure
each. The results are reported below in millimeters.
42 39 42 45 43 40 39 41 40 42
43 42 45 47 38 46 48 42 47 41
40 44 43 42 40 41 42 39 40 41
Is it reasonable to conclude that the mean length of the bars reduced?
Answer by CPhill(2264) (Show Source): You can put this solution on YOUR website!
```python?code_reference&code_event_index=3
import numpy as np
from scipy import stats
data = [
42, 39, 42, 45, 43, 40, 39, 41, 40, 42,
43, 42, 45, 47, 38, 46, 48, 42, 47, 41,
40, 44, 43, 42, 40, 41, 42, 39, 40, 41
]
mu_0 = 43
n = len(data)
x_bar = np.mean(data)
s = np.std(data, ddof=1)
se = s / np.sqrt(n)
t_stat = (x_bar - mu_0) / se
df = n - 1
# One-tailed test (Reduced mean: HA: mu < 43)
p_value = stats.t.cdf(t_stat, df)
print(f"{x_bar=}")
print(f"{s=}")
print(f"{t_stat=}")
print(f"{p_value=}")
```
```text?code_stdout&code_event_index=3
x_bar=42.13333333333333
s=2.582879085016581
t_stat=-1.8378440010020713
p_value=0.038175278168928364
```
To determine if the mean length of the bars has reduced, we conduct a **one-sample t-test**.
### 1. State the Hypotheses
* **Null Hypothesis ($H_0$):** $\mu \geq 43$ (The mean length has not reduced)
* **Alternative Hypothesis ($H_A$):** $\mu < 43$ (The mean length has reduced)
### 2. Identify Sample Statistics
From the provided data of $n = 30$ bars:
* **Sample Mean ($\bar{x}$):** $42.13$ mm
* **Sample Standard Deviation ($s$):** $2.58$ mm
* **Hypothesized Mean ($\mu_0$):** $43$ mm
### 3. Calculate the Test Statistic ($t$)
The $t$-statistic measures how many standard errors the sample mean is from the hypothesized mean:
$$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} = \frac{42.13 - 43}{2.58 / \sqrt{30}} \approx -1.838$$
### 4. Determine the p-value
Using the $t$-distribution with $df = n - 1 = 29$:
* **$p$-value:** $\approx 0.0382$
### 5. Conclusion
At a standard significance level of **$\alpha = 0.05$**:
* Since the $p$-value ($0.0382$) is **less than** $0.05$, we **reject the null hypothesis**.
**Final Result:** Yes, it is **reasonable to conclude** that the mean length of the bars has reduced. The statistical evidence indicates that the decrease from $43$ mm to $42.13$ mm is significant enough that it is unlikely to have occurred by random chance alone.
> **Note:** If a more stringent significance level like $\alpha = 0.01$ were used, we would fail to reject the null hypothesis, as $0.0382 > 0.01$. However, in most industrial engineering contexts, $0.05$ is the standard threshold for investigation.
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