t-test question

questions/186_one-sample-z-test-hypothesis-testing/description.md

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+Implement a function to perform a one-sample Z-test for a population mean when the population standard deviation is known. Your function must support both one-tailed and two-tailed alternatives.
+
+Implement a function with the signature:
+- one_sample_z_test(sample_mean, population_mean, population_std, n, alternative="two-sided")
+
+Where:
+- sample_mean: The observed sample mean (float)
+- population_mean: The hypothesized population mean under H0 (float)
+- population_std: The known population standard deviation (float > 0)
+- n: Sample size (int > 0)
+- alternative: One of {"two-sided", "greater", "less"}
+
+Return a dictionary with:
+- "z": the computed Z statistic rounded to 4 decimals
+- "p_value": the corresponding p-value rounded to 4 decimals
+
+Use the standard normal distribution for the p-value. Handle invalid inputs minimally by assuming valid types and values.
+

questions/186_one-sample-z-test-hypothesis-testing/example.json

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+{
+  "input": "sample_mean=103.0, population_mean=100.0, population_std=15.0, n=36, alternative='greater'",
+  "output": "{'z': 1.2, 'p_value': 0.1151}",
+  "reasoning": "Standard error = 15/sqrt(36)=2.5. Z=(103-100)/2.5=1.2. For a 'greater' test, p=1-CDF(1.2)=0.1151."
+}
+

questions/186_one-sample-z-test-hypothesis-testing/learn.md

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+A one-sample Z-test assesses whether the mean of a population differs from a hypothesized value when the population standard deviation is known. It is appropriate for large samples (by CLT) or when normality is assumed and the population standard deviation is known.
+
+Test statistic:
+- z = (x̄ − μ0) / (σ / √n)
+  - x̄: sample mean
+  - μ0: hypothesized mean under H0
+  - σ: known population standard deviation
+  - n: sample size
+
+P-value computation uses the standard normal distribution:
+- Two-sided (H1: μ ≠ μ0): p = 2 · min(Φ(z), 1 − Φ(z))
+- Right-tailed (H1: μ > μ0): p = 1 − Φ(z)
+- Left-tailed (H1: μ < μ0): p = Φ(z)
+
+Decision at level α:
+- Reject H0 if p ≤ α; otherwise, fail to reject H0.
+
+Notes:
+- If σ is unknown, use a one-sample t-test with the sample standard deviation instead.
+

questions/186_one-sample-z-test-hypothesis-testing/meta.json

@@ -0,0 +1,13 @@
+{
+  "id": "186",
+  "title": "One-Sample Z-Test for Mean (One and Two-Tailed)",
+  "difficulty": "easy",
+  "category": "Statistics",
+  "video": "",
+  "likes": "0",
+  "dislikes": "0",
+  "contributor": [
+    { "profile_link": "https://github.com/Jeet009", "name": "Jeet Mukherjee" }
+  ]
+}
+

questions/186_one-sample-z-test-hypothesis-testing/solution.py

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+from math import erf, sqrt
+
+def _standard_normal_cdf(x):
+    return 0.5 * (1.0 + erf(x / sqrt(2.0)))
+
+def one_sample_z_test(sample_mean, population_mean, population_std, n, alternative="two-sided"):
+    """
+    Perform a one-sample Z-test for a population mean with known population std.
+
+    Parameters
+    ----------
+    sample_mean : float
+    population_mean : float
+    population_std : float
+    n : int
+    alternative : str
+        One of {"two-sided", "greater", "less"}
+
+    Returns
+    -------
+    dict with keys:
+      - "z": Z-statistic rounded to 4 decimals
+      - "p_value": p-value rounded to 4 decimals
+    """
+    standard_error = population_std / sqrt(n)
+    z = (sample_mean - population_mean) / standard_error
+    cdf = _standard_normal_cdf(z)
+
+    if alternative == "two-sided":
+        p = 2.0 * min(cdf, 1.0 - cdf)
+    elif alternative == "greater":
+        p = 1.0 - cdf
+    elif alternative == "less":
+        p = cdf
+    else:
+        # Fallback to two-sided if unexpected input
+        p = 2.0 * min(cdf, 1.0 - cdf)
+
+    return {"z": round(z, 4), "p_value": round(p, 4)}
+

questions/186_one-sample-z-test-hypothesis-testing/starter_code.py

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+from math import erf, sqrt
+
+def _standard_normal_cdf(x):
+    return 0.5 * (1.0 + erf(x / sqrt(2.0)))
+
+def one_sample_z_test(sample_mean, population_mean, population_std, n, alternative="two-sided"):
+    """
+    Perform a one-sample Z-test for a population mean with known population std.
+
+    Parameters
+    ----------
+    sample_mean : float
+    population_mean : float
+    population_std : float
+    n : int
+    alternative : str
+        One of {"two-sided", "greater", "less"}
+
+    Returns
+    -------
+    dict with keys:
+      - "z": Z-statistic rounded to 4 decimals
+      - "p_value": p-value rounded to 4 decimals
+    """
+    # TODO: Implement the Z statistic and p-value computation
+    # z = (sample_mean - population_mean) / (population_std / sqrt(n))
+    # Use _standard_normal_cdf for CDF of standard normal.
+    # For alternative:
+    # - "two-sided": p = 2 * min(P(Z<=z), P(Z>=z)) = 2 * min(cdf(z), 1-cdf(z))
+    # - "greater":   p = 1 - cdf(z)
+    # - "less":      p = cdf(z)
+    return {"z": 0.0, "p_value": 1.0}
+

questions/186_one-sample-z-test-hypothesis-testing/tests.json

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+[
+  {
+    "test": "one_sample_z_test(103.0, 100.0, 15.0, 36, alternative='two-sided')",
+    "expected_output": "{'z': 1.2, 'p_value': 0.2296}"
+  },
+  {
+    "test": "one_sample_z_test(103.0, 100.0, 15.0, 36, alternative='greater')",
+    "expected_output": "{'z': 1.2, 'p_value': 0.1151}"
+  },
+  {
+    "test": "one_sample_z_test(103.0, 100.0, 15.0, 36, alternative='less')",
+    "expected_output": "{'z': 1.2, 'p_value': 0.8849}"
+  },
+  {
+    "test": "one_sample_z_test(97.0, 100.0, 10.0, 25, alternative='two-sided')",
+    "expected_output": "{'z': -1.5, 'p_value': 0.1336}"
+  },
+  {
+    "test": "one_sample_z_test(97.0, 100.0, 10.0, 25, alternative='less')",
+    "expected_output": "{'z': -1.5, 'p_value': 0.0668}"
+  },
+  {
+    "test": "one_sample_z_test(97.0, 100.0, 10.0, 25, alternative='greater')",
+    "expected_output": "{'z': -1.5, 'p_value': 0.9332}"
+  }
+]
+

questions/187_one-sample-t-test-hypothesis-testing/description.md

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+Implement a function to perform a one-sample t-test for a population mean when the population standard deviation is unknown (use sample standard deviation). Your function must auto-detect whether the test is one-tailed or two-tailed by parsing the hypotheses.
+
+Implement a function with the signature:
+- one_sample_t_test(sample_mean, sample_std, n, H0, H1)
+
+Where:
+- sample_mean: Observed sample mean (float)
+- sample_std: Sample standard deviation (float > 0, computed with ddof=1)
+- n: Sample size (int > 1)
+- H0: Null hypothesis as a string, e.g., "mu = 100"
+- H1: Alternative hypothesis as a string, e.g., "mu > 100", "mu < 100", or "mu != 100"
+
+Requirements:
+- Extract the hypothesized mean μ0 from H0.
+- Determine the tail from H1:
+  - "mu != μ0" → two-sided
+  - "mu > μ0" → right-tailed
+  - "mu < μ0" → left-tailed
+- Compute the t-statistic: t = (x̄ − μ0) / (s / √n)
+- Degrees of freedom: df = n − 1
+- Compute the p-value using the Student's t distribution.
+
+Return a dictionary with:
+- "t": computed t-statistic rounded to 4 decimals
+- "df": degrees of freedom (int)
+- "p_value": p-value rounded to 4 decimals
+- "alternative": one of {"two-sided", "greater", "less"}
+

questions/187_one-sample-t-test-hypothesis-testing/example.json

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+{
+  "input": "sample_mean=103.0, sample_std=10.0, n=25, H0='mu = 100', H1='mu > 100'",
+  "output": "{'t': 1.5, 'df': 24, 'p_value': 0.0735, 'alternative': 'greater'}",
+  "reasoning": "SE = 10/sqrt(25)=2.0. t=(103-100)/2=1.5. With df=24 and a right-tailed test, p=1-CDF_t(1.5;24)=0.0735 (approx)."
+}
+

questions/187_one-sample-t-test-hypothesis-testing/learn.md

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+A one-sample t-test assesses whether a population mean differs from a hypothesized value when the population standard deviation is unknown. It uses the sample standard deviation and Student's t distribution with df = n − 1.
+
+Test statistic:
+- t = (x̄ − μ0) / (s / √n)
+  - x̄: sample mean
+  - μ0: hypothesized mean under H0
+  - s: sample standard deviation (ddof = 1)
+  - n: sample size
+
+Tail selection from hypotheses:
+- If H1 is "mu != μ0", it's two-sided: p = 2 · min(Tcdf(t), 1 − Tcdf(t))
+- If H1 is "mu > μ0", it's right-tailed: p = 1 − Tcdf(t)
+- If H1 is "mu < μ0", it's left-tailed: p = Tcdf(t)
+
+Decision rule at level α:
+- Reject H0 if p ≤ α; otherwise, fail to reject H0.
+
+When to use:
+- Use the t-test when σ is unknown and the sample is reasonably normal or n is moderate/large.
+

questions/187_mlops-etl-pipeline/meta.json → questions/187_one-sample-t-test-hypothesis-testing/meta.json

@@ -1,12 +1,13 @@
 {
   "id": "187",
-  "title": "Build a Simple ETL Pipeline (MLOps)",
-  "difficulty": "medium",
-  "category": "MLOps",
+  "title": "One-Sample t-Test for Mean (One and Two-Tailed)",
+  "difficulty": "easy",
+  "category": "Statistics",
   "video": "",
   "likes": "0",
   "dislikes": "0",
   "contributor": [
     { "profile_link": "https://github.com/Jeet009", "name": "Jeet Mukherjee" }
   ]
 }
+