Statistical inference

A test of $H_0$: "the new fertiliser has no effect on mean yield" gives $p = 0.03$. Which statement is correct?

A health clinic records the number of patients arriving in five consecutive hours: $3, 5, 2, 6, 4$. Modelling the counts as independent Poisson$(\lambda)$, what is the maximum likelihood estimate of $\lambda$?

Compute the sample variance $S^2 = \frac{1}{n-1}\sum_i (x_i - \bar x)^2$ of the data $2, 4, 9$.

For an i.i.d. sample $X_1, \dots, X_n$ from the uniform distribution on $[0, \theta]$, the MLE is $M = \max_i X_i$, and $E[M] = \frac{n}{n+1}\theta$. Which estimator is unbiased for $\theta$?

In the proof that the normal equations give the least-squares solution, we expand $\|y - X\beta\|^2 = \|r\|^2 + 2\,r^\top Xw + \|Xw\|^2$ with $r = y - X\hat\beta$ and $w = \hat\beta - \beta$. Why does the cross term vanish?

A study tests $8$ hypotheses and obtains the p-values $0.0004,\ 0.003,\ 0.006,\ 0.0071,\ 0.011,\ 0.02,\ 0.04,\ 0.3$. Using the Bonferroni correction with family-wise error rate $0.05$, how many hypotheses are rejected?

A small trial of a new treatment (20 patients) finds a $95\%$ confidence interval of $[-2, 15]$ days for the reduction in recovery time, with $p = 0.13$. What is the best conclusion?

From $260$ of $400$ households, a $95\%$ confidence interval for the proportion $p$ is $[0.603, 0.697]$, and with a uniform prior the Bayesian posterior is Beta$(261, 141)$ with $95\%$ credible interval $[0.602, 0.695]$. Which statements are correct? (Choose all that apply.)

Which Lean statement says that a solution b of the normal equations minimises the residual sum of squares? (Xᵀ is the transpose, *ᵥ multiplies a matrix by a vector, ⬝ᵥ is the dot product.)

Write fit_line(xs, ys) returning the least-squares (intercept, slope) for the model $y = \beta_0 + \beta_1 x$, using $\hat\beta_1 = \frac{\sum (x_i - \bar x)(y_i - \bar y)}{\sum (x_i - \bar x)^2}$ and $\hat\beta_0 = \bar y - \hat\beta_1\bar x$. Use Fraction so the answers are exact. The tests also check that the residuals sum to zero and are orthogonal to $x$ — the normal equations.