Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =4\). For \(\displaystyle n=25\), test \(H_0:\mu=61\) against \(H_1:\mu>61\). The sample mean is \(62.92\).

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

Find the observed z statistic.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =5\). For \(\displaystyle n=25\), test \(H_0:\mu=62\) against \(H_1:\mu>62\). The sample mean is \(64.6\).

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

Find the observed z statistic.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =5\). For \(\displaystyle n=25\), test \(H_0:\mu=62\) against \(H_1:\mu>62\). The sample mean is \(64.6\).

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

Find the right-sided p-value.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =6\). For \(\displaystyle n=25\), test \(H_0:\mu=63\) against \(H_1:\mu>63\). The sample mean is \(66.36\).

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

Find the right-sided p-value.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =6\). For \(\displaystyle n=25\), test \(H_0:\mu=63\) against \(H_1:\mu>63\). The sample mean is \(66.36\).

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

Find the sample-mean critical boundary at 5% significance.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =7\). For \(\displaystyle n=25\), test \(H_0:\mu=64\) against \(H_1:\mu>64\). The sample mean is \(66.24\).

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

Find the sample-mean critical boundary at 5% significance.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =7\). For \(\displaystyle n=25\), test \(H_0:\mu=64\) against \(H_1:\mu>64\). The sample mean is \(66.24\).

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

Find the two-sided p-value for the same observed z.

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

Maths4U Original · Short practice

Observations are independent and normal with known \(\displaystyle \sigma =3\). For \(\displaystyle n=25\), test \(H_0:\mu=65\) against \(H_1:\mu>65\). The sample mean is \(66.08\).

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

Find the two-sided p-value for the same observed z.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(14,p)\). Test \(H_0:p=0.2\) against a larger p, rejecting for \(X\ge 6\). For power use \(p=0.4\). The observed count is 7.

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

Find the actual significance of this critical region.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(15,p)\). Test \(H_0:p=0.25\) against a larger p, rejecting for \(X\ge 8\). For power use \(p=0.45\). The observed count is 9.

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

Find the actual significance of this critical region.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(15,p)\). Test \(H_0:p=0.25\) against a larger p, rejecting for \(X\ge 8\). For power use \(p=0.45\). The observed count is 9.

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

Find the power at the specified alternative.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(16,p)\). Test \(H_0:p=0.3\) against a larger p, rejecting for \(X\ge 9\). For power use \(p=0.5\). The observed count is 10.

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

Find the power at the specified alternative.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(16,p)\). Test \(H_0:p=0.3\) against a larger p, rejecting for \(X\ge 9\). For power use \(p=0.5\). The observed count is 10.

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

Find the Type II error probability at that alternative.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(12,p)\). Test \(H_0:p=0.2\) against a larger p, rejecting for \(X\ge 6\). For power use \(p=0.4\). The observed count is 7.

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

Find the Type II error probability at that alternative.

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(12,p)\). Test \(H_0:p=0.2\) against a larger p, rejecting for \(X\ge 6\). For power use \(p=0.4\). The observed count is 7.

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

Find the one-sided p-value when \(\displaystyle X=7\).

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

Maths4U Original · Short practice

\(X\sim\operatorname{Bin}(13,p)\). Test \(H_0:p=0.25\) against a larger p, rejecting for \(X\ge 7\). For power use \(p=0.45\). The observed count is 8.

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

Find the one-sided p-value when \(\displaystyle X=8\).

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(3.25\) against an increase, using critical region \(X\ge 7\). For the alternative calculation the mean is \(5.25\). The observed count is 7.

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

Find the actual significance.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(3.5\) against an increase, using critical region \(X\ge 8\). For the alternative calculation the mean is \(5.5\). The observed count is 8.

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

Find the actual significance.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(3.5\) against an increase, using critical region \(X\ge 8\). For the alternative calculation the mean is \(5.5\). The observed count is 8.

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

Find the power at the specified alternative.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(3.75\) against an increase, using critical region \(X\ge 8\). For the alternative calculation the mean is \(5.75\). The observed count is 8.

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

Find the power at the specified alternative.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(3.75\) against an increase, using critical region \(X\ge 8\). For the alternative calculation the mean is \(5.75\). The observed count is 8.

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

Find the Type II probability at that alternative.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(4\) against an increase, using critical region \(X\ge 9\). For the alternative calculation the mean is \(6\). The observed count is 9.

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

Find the Type II probability at that alternative.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(4\) against an increase, using critical region \(X\ge 9\). For the alternative calculation the mean is \(6\). The observed count is 9.

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

Find the null probability of no events.

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

Maths4U Original · Short practice

During a fixed interval, X has a Poisson distribution. Test null mean \(4.25\) against an increase, using critical region \(X\ge 9\). For the alternative calculation the mean is \(6.25\). The observed count is 9.

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

Find the null probability of no events.

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