Statistical Tests and Distributions Questions and Answers

At a school the pass rate in 'A' level mathematics was $40\%$. After a new teacher was hired, $4$ out of $9$ students passed. A test is to be run at the $5\%$ level for evidence of an improvement. Which pair of hypotheses is correct?

The number of passes $X$ among $9$ students is modelled by $X \sim B(9, 0.4)$. Find $P(X \ge 4)$, to 3 decimal places.

A one-tailed test at the $5\%$ level uses $H_0: p = 0.4$ against $H_1: p > 0.4$, with $4$ passes out of $9$ observed and $P(X \ge 4) = 0.517$ under $H_0$. What is the correct conclusion?

The distance travelled by a commuter driver in a day is normally distributed with mean $360$ km and standard deviation $60$ km. Find the probability that the distance travelled in a day exceeds $370$ km, to 3 decimal places.

A confidence interval for a population mean is to be given at the $98\%$ level, using a normal distribution. State the value of $z$ used, to 3 decimal places.

A random sample of $30$ days gives a mean distance of $350$ km, where the population standard deviation is known to be $60$ km. Which is the $98\%$ confidence interval for the mean distance travelled, to 1 decimal place?