DP Mathematics HL Questionbank

• Syllabus

7.3

Sub sections and their related questions

Unbiased estimators and estimates.

• 12M.3sp.hl.TZ0.1a: Determine unbiased estimates for the mean and variance of the distribution.
• 11M.3sp.hl.TZ0.3a: Determine unbiased estimates of the mean and variance of the loss in weight achieved over the...
• SPNone.3sp.hl.TZ0.1a: Determine unbiased estimates of $\mu$ and ${\sigma ^2}$.
• SPNone.3sp.hl.TZ0.5b: In order to estimate $\theta$, a random sample of n observations is obtained from the...
• 10N.2.hl.TZ0.2: The company Fresh Water produces one-litre bottles of mineral water. The company wants to...
• 13M.3sp.hl.TZ0.1a: Find unbiased estimates of $\mu$ and ${\sigma ^2}$.
• 14M.3sp.hl.TZ0.3: (a)     Consider the random variable $X$ for which ${\text{E}}(X) = a\lambda  + b$, where...
• 14N.3sp.hl.TZ0.4c: Let $T = \frac{Y}{2} + \frac{Y}{2}$. (i)     Show that $T$ is an unbiased estimator for...
• 15M.3sp.hl.TZ0.4a: Explain briefly the meaning of (i)     an estimator of $\mu$; (ii)     an unbiased estimator...
• 15M.3sp.hl.TZ0.4b: A random sample ${X_1},{\text{ }}{X_2},{\text{ }}{X_3}$ of three independent observations is...
• 15N.3sp.hl.TZ0.5a: (i)     Find ${\text{E}}({X_1})$. (ii)     Hence obtain an unbiased estimator for $p$.
• 15N.3sp.hl.TZ0.5b: The die is rolled a second time, and the score ${X_2}$ is noted. (i)     Show that...
• 18M.3sp.hl.TZ0.5a: Show that $P = \frac{X}{n}$ is an unbiased estimator of $p$.
• 18M.3sp.hl.TZ0.5b.i: Show that ${\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)$.
• 18M.3sp.hl.TZ0.5b.ii: Hence write down an unbiased estimator of Var(X).

Comparison of unbiased estimators based on variances.

• 14N.3sp.hl.TZ0.4c: Let $T = \frac{Y}{2} + \frac{Y}{2}$. (i)     Show that $T$ is an unbiased estimator for...
• 15M.3sp.hl.TZ0.4b: A random sample ${X_1},{\text{ }}{X_2},{\text{ }}{X_3}$ of three independent observations is...
• 15N.3sp.hl.TZ0.5b: The die is rolled a second time, and the score ${X_2}$ is noted. (i)     Show that...
• 18M.3sp.hl.TZ0.5a: Show that $P = \frac{X}{n}$ is an unbiased estimator of $p$.
• 18M.3sp.hl.TZ0.5b.i: Show that ${\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)$.
• 18M.3sp.hl.TZ0.5b.ii: Hence write down an unbiased estimator of Var(X).

${\bar X}$ as an unbiased estimator for $\mu$ .

• 15M.3sp.hl.TZ0.2a: Determine unbiased estimates of $\mu$ and ${\sigma ^2}$.
• 18M.3sp.hl.TZ0.5a: Show that $P = \frac{X}{n}$ is an unbiased estimator of $p$.
• 18M.3sp.hl.TZ0.5b.i: Show that ${\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)$.
• 18M.3sp.hl.TZ0.5b.ii: Hence write down an unbiased estimator of Var(X).

${S^2}$ as an unbiased estimator for ${\sigma ^2}$ .

• 15M.3sp.hl.TZ0.2a: Determine unbiased estimates of $\mu$ and ${\sigma ^2}$.
• 18M.3sp.hl.TZ0.5a: Show that $P = \frac{X}{n}$ is an unbiased estimator of $p$.
• 18M.3sp.hl.TZ0.5b.i: Show that ${\text{E}}\left( U \right) = \left( {n - 1} \right)p\left( {1 - p} \right)$.
• 18M.3sp.hl.TZ0.5b.ii: Hence write down an unbiased estimator of Var(X).