Cross-topic review exercise 2 · Short practice

Recall

Transformations of continuous variables — objective 1

Maths4U Original · Short practice

X has density \(\displaystyle \frac{2x}{121}\) on \(0\le x\le 11\). Define \(\displaystyle Y=X^{2}\). For \(0\le y\le 121\), \(\displaystyle F_{Y}(y)= P(X\le \sqrt{y})= \frac{y}{121}\).

Find the constant density of Y on its support.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Transformations of continuous variables — objective 1

Maths4U Original · Short practice

X has density \(\displaystyle \frac{2x}{144}\) on \(0\le x\le 12\). Define \(\displaystyle Y=X^{2}\). For \(0\le y\le 144\), \(\displaystyle F_{Y}(y)= P(X\le \sqrt{y})= \frac{y}{144}\).

Find the constant density of Y on its support.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Method

Two-sample pooled t inference — objective 2

Maths4U Original · Short practice

Two independent normal populations have a common unknown variance. Sample sizes are 9 and 8, their unbiased variances are 6 and 6, and \(\displaystyle \bar x_1-\bar x_2=\frac{3}{2}\).

Use \(\displaystyle H_{1}: \mu _{1}>\mu _{2}\) and a 5% significance level for the written test conclusion.

Find the standard error of the mean difference.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Two-sample pooled t inference — objective 2

Maths4U Original · Short practice

Two independent normal populations have a common unknown variance. Sample sizes are 10 and 9, their unbiased variances are 7 and 7, and \(\bar x_1-\bar x_2=2\).

Use \(\displaystyle H_{1}: \mu _{1}>\mu _{2}\) and a 5% significance level for the written test conclusion.

Find the standard error of the mean difference.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Apply

Chi-squared tests of independence — objective 3

Maths4U Original · Short practice

Independent observations form the following two-by-three table. Test independence without a continuity correction.

Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.

Find the p-value.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

Chi-squared tests of independence — objective 3

Maths4U Original · Short practice

Independent observations form the following two-by-three table. Test independence without a continuity correction.

Use a 5% significance level for a written conclusion. A distribution calculator may be used for numerical p-values.

Find the p-value.

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Check understanding

PGFs of independent sums — objective 4

Maths4U Original · Short practice

\(\displaystyle X_{1}\),…,\(\displaystyle X_{m}\) are independent \(\displaystyle \operatorname{Bin}(3,\frac{2}{5})\) variables, with \(\displaystyle m=2\). Let S be their sum. Thus \(G_S(t)=(1-p+pt)^{6}\).

Find \(\displaystyle P(S=1)\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.

Fresh review variant

PGFs of independent sums — objective 4

Maths4U Original · Short practice

\(\displaystyle X_{1}\),…,\(\displaystyle X_{m}\) are independent \(\displaystyle \operatorname{Bin}(4,\frac{1}{2})\) variables, with \(\displaystyle m=3\). Let S be their sum. Thus \(G_S(t)=(1-p+pt)^{12}\).

Find \(\displaystyle P(S=1)\).

Enter one number or fraction. Use at least six decimal places when needed; a distribution calculator may be used for probabilities.