Mathematics 9709 · AS & A Level · Hypothesis testing

Hypothesis testing — practice question

During the previous year, the average amount of time that students at a school needed to finish a particular test was $25$ minutes. Akash thinks that the average time needed by this year’s students is below $25$ minutes. To check this idea, he selects a large random sample of this year’s students and records the time for each student. He carries out a test, at the $2.5\%$ significance level, for the population mean time, $\mu$ minutes. Akash uses the null hypothesis $H_0: \mu = 25$.
(a)[1]

Give a reason why Akash ought to use a one-tailed test.

(b)[2]

Akash obtains a value of the test statistic of $z = -2.02$. Explain what conclusion he should draw.

(c)[3]

In another one-tailed hypothesis test, the $z$-value was found to be $2.14$. If this value would result in the null hypothesis being rejected at the $\alpha\%$ significance level, find the range of possible values of $\alpha$.

(d)[3]

The population mean time taken by students at another school to complete a test last year was $m$ minutes. Sorin carries out a one-tailed test to decide whether the population mean this year is less than $m$, using a random sample of $100$ students. He takes the population standard deviation of the times to be $3.9$ minutes. The sample mean is $24.8$ minutes, and this outcome only just causes rejection of the null hypothesis at the $5\%$ significance level. Find the value of $m$.

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