5th semester files

5th-Semester-Fall-2023/Prob-and-Stat-for-ECEs/Homework/Homework-10.md

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+# Homework 10 - Aidan Sharpe
+
+## 1
+The titanium content in an aircraft-grade alloy is an important discriminant of strength. A sample of 10 test coupns reveals the following titanium content in percent:
+
+| Coupon | Titanium Content |
+| ------ | ---------------- |
+| 1      | 8.30%            |
+| 2      | 8.09%            |
+| 3      | 8.99%            |
+| 4      | 8.60%            |
+| 5      | 8.40%            |
+| 6      | 8.35%            |
+| 7      | 8.36%            |
+| 8      | 8.75%            |
+| 9      | 8.91%            |
+| 10     | 8.05%            |
+
+Suppose that the distribution of titanium content is symmetric and continuous. Does the sample data suggest that the mean titanium content differs significantly from 8.5%? Use $\alpha = 0.05$
+
+```python
+>>> coupons = [8.3, 8.09, 8.99, 8.60, 8.40, 8.35, 8.36, 8.75, 8.91, 8.05]
+>>> mu0 = 8.5
+
+# Find the difference between each sample and mu0
+>>> differences = [xi - mu0 for xi in coupons]
+
+# Find the absolute differences
+>>> abs_diffs = [abs(x) for x in differences]
+
+# Sort the differences in ascending order
+>>> s_diffs = sorted(abs_diffs)
+
+# Turn the sorted order into a ranked list
+>>> ranks = [abs_diffs.index(x) + 1 for x in s_diffs]
+
+# Find the ranks corresponding to positive differences
+>>> p_ranks = [ranks[i] if differences[i] > 0 else 0 for i in range(len(differences))]
+
+# Find the ranks corresponding to negative differences
+>>> n_ranks = [ranks[i] if differences[i] < 0 else 0 for i in range(len(differences))]
+
+# Find the positive and negative rank sums
+>>> wp = sum(p_ranks)
+>>> wn = sum(n_ranks)
+
+# Find the test statistic
+>>> w_observed = min(wp, wn)
+>>> w_observed
+22
+```
+
+For a two-sided signed rank test with $\alpha = 0.05$ and 10 samples, $w_\alpha^*$ is 8. Since 22 is greater than 8, we do not have enough evidence to suggest that the mean titanium content differs from 8.5%.