added t-test question

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_one-sample-t-test-hypothesis-testing/meta.json

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+{
+  "id": "187",
+  "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" }
+  ]
+}
+

questions/187_one-sample-t-test-hypothesis-testing/solution.py

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+from math import gamma, sqrt, pi
+
+def _t_pdf(x, df):
+    c = gamma((df + 1) / 2.0) / (sqrt(df * pi) * gamma(df / 2.0))
+    return c * (1.0 + (x * x) / df) ** (-(df + 1) / 2.0)
+
+def _t_cdf(x, df):
+    if x == 0.0:
+        return 0.5
+    sign = 1.0 if x > 0 else -1.0
+    a, b = 0.0, abs(x)
+    n = 4000  # even number of intervals; higher for better accuracy
+    h = (b - a) / n
+    s = _t_pdf(a, df) + _t_pdf(b, df)
+    for i in range(1, n):
+        xi = a + i * h
+        s += (4 if i % 2 == 1 else 2) * _t_pdf(xi, df)
+    integral = s * h / 3.0
+    return 0.5 + sign * integral
+
+def _parse_hypotheses(H0, H1):
+    text = H0.replace(" ", "")
+    if "=" not in text:
+        raise ValueError("H0 must specify equality, e.g., 'mu = 100'")
+    mu0 = float(text.split("=")[1])
+    alt_text = H1.replace(" ", "").replace("<>", "!=").replace("≠", "!=")
+    if "!=" in alt_text:
+        alt = "two-sided"
+    elif ">" in alt_text:
+        alt = "greater"
+    elif "<" in alt_text:
+        alt = "less"
+    else:
+        alt = "two-sided"
+    return mu0, alt
+
+def one_sample_t_test(sample_mean, sample_std, n, H0, H1):
+    """
+    Perform a one-sample t-test for a population mean with unknown population std.
+    Auto-detect tail from H1 and extract μ0 from H0.
+    Returns dict with keys: "t", "df", "p_value", "alternative".
+    """
+    mu0, alternative = _parse_hypotheses(H0, H1)
+    df = n - 1
+    se = sample_std / sqrt(n)
+    t = (sample_mean - mu0) / se
+    cdf = _t_cdf(t, df)
+
+    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:
+        p = 2.0 * min(cdf, 1.0 - cdf)
+
+    return {"t": round(t, 4), "df": df, "p_value": round(p, 4), "alternative": alternative}
+

questions/187_one-sample-t-test-hypothesis-testing/starter_code.py

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+from math import gamma, sqrt, pi
+
+def _t_pdf(x, df):
+    c = gamma((df + 1) / 2.0) / (sqrt(df * pi) * gamma(df / 2.0))
+    return c * (1.0 + (x * x) / df) ** (-(df + 1) / 2.0)
+
+def _t_cdf(x, df):
+    # Numerical integration using Simpson's rule, leveraging symmetry
+    if x == 0.0:
+        return 0.5
+    sign = 1.0 if x > 0 else -1.0
+    a, b = 0.0, abs(x)
+    n = 2000  # even number of intervals
+    h = (b - a) / n
+    s = _t_pdf(a, df) + _t_pdf(b, df)
+    for i in range(1, n):
+        xi = a + i * h
+        s += (4 if i % 2 == 1 else 2) * _t_pdf(xi, df)
+    integral = s * h / 3.0
+    return 0.5 + sign * integral
+
+def _parse_hypotheses(H0, H1):
+    # Extract mu0 from H0 like "mu = 100" (allow spaces)
+    text = H0.replace(" ", "")
+    if "=" not in text:
+        raise ValueError("H0 must specify equality, e.g., 'mu = 100'")
+    mu0 = float(text.split("=")[1])
+    alt_text = H1.replace(" ", "").replace("<>", "!=").replace("≠", "!=")
+    if "!=" in alt_text:
+        alt = "two-sided"
+    elif ">" in alt_text:
+        alt = "greater"
+    elif "<" in alt_text:
+        alt = "less"
+    else:
+        alt = "two-sided"
+    return mu0, alt
+
+def one_sample_t_test(sample_mean, sample_std, n, H0, H1):
+    """
+    Perform a one-sample t-test for a population mean with unknown population std.
+    Auto-detect tail from H1 and extract μ0 from H0.
+    Returns dict with keys: "t", "df", "p_value", "alternative".
+    """
+    # TODO: implement
+    # Steps:
+    # 1) mu0, alternative = _parse_hypotheses(H0, H1)
+    # 2) df = n - 1
+    # 3) t = (sample_mean - mu0) / (sample_std / sqrt(n))
+    # 4) p-value using _t_cdf:
+    #    - two-sided: p = 2 * min(CDF(t), 1 - CDF(t))
+    #    - greater:   p = 1 - CDF(t)
+    #    - less:      p = CDF(t)
+    return {"t": 0.0, "df": n - 1, "p_value": 1.0, "alternative": "two-sided"}
+

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

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+[
+  {
+    "test": "one_sample_t_test(103.0, 10.0, 25, 'mu = 100', 'mu != 100')",
+    "expected_output": "{'t': 1.5, 'df': 24, 'p_value': 0.1469, 'alternative': 'two-sided'}"
+  },
+  {
+    "test": "one_sample_t_test(103.0, 10.0, 25, 'mu = 100', 'mu > 100')",
+    "expected_output": "{'t': 1.5, 'df': 24, 'p_value': 0.0735, 'alternative': 'greater'}"
+  },
+  {
+    "test": "one_sample_t_test(103.0, 10.0, 25, 'mu = 100', 'mu < 100')",
+    "expected_output": "{'t': 1.5, 'df': 24, 'p_value': 0.9265, 'alternative': 'less'}"
+  },
+  {
+    "test": "one_sample_t_test(97.0, 12.0, 36, 'mu = 100', 'mu != 100')",
+    "expected_output": "{'t': -1.5, 'df': 35, 'p_value': 0.1423, 'alternative': 'two-sided'}"
+  },
+  {
+    "test": "one_sample_t_test(97.0, 12.0, 36, 'mu = 100', 'mu < 100')",
+    "expected_output": "{'t': -1.5, 'df': 35, 'p_value': 0.0712, 'alternative': 'less'}"
+  },
+  {
+    "test": "one_sample_t_test(97.0, 12.0, 36, 'mu = 100', 'mu > 100')",
+    "expected_output": "{'t': -1.5, 'df': 35, 'p_value': 0.9288, 'alternative': 'greater'}"
+  }
+]
+