State probability of Type I error

5 The time taken for a particular train journey is normally distributed. In the past, the time had mean 2.4 hours and standard deviation 0.3 hours. A new timetable is introduced and on 30 randomly chosen occasions the time for this journey is measured. The mean time for these 30 occasions is found to be 2.3 hours.

1. Stating any assumption(s), test, at the \(5 \%\) significance level, whether the mean time for this journey has changed.
2. A similar test at the \(5 \%\) significance level was carried out using the times from another randomly chosen 30 occasions.
   1. State the probability of a Type I error.
   2. State what is meant by a Type II error in this context.

7 The heights of one-year-old trees of a certain variety are known to have mean 2.3 m . A scientist believes that, on average, trees of this age and variety in her region are slightly taller than in other places. She plans to carry out a hypothesis test, at the \(2 \%\) significance level, in order to test her belief.

1. State the probability that she will make a Type I error.
   She takes a random sample of 100 such trees in her region and measures their heights, \(h \mathrm {~m}\). Her results are summarised below. $$n = 100 \quad \sum h = 238 \quad \sum h ^ { 2 } = 580$$
2. Carry out the test at the \(2 \%\) significance level. \includegraphics[max width=\textwidth, alt={}, center]{9ac74d4c-f5e0-4c5d-ab25-5692dfb06f0b-10_2717_35_109_2012}
3. The scientist carries out the test correctly, but another scientist claims that she has made a Type II error. Comment on this claim.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The heights, in centimetres, of adult females in Litania have mean \(\mu\) and standard deviation \(\sigma\). It is known that in 2004 the values of \(\mu\) and \(\sigma\) were 163.21 and 6.95 respectively. The government claims that the value of \(\mu\) this year is greater than it was in 2004. In order to test this claim a researcher plans to carry out a hypothesis test at the \(1 \%\) significance level. He records the heights of a random sample of 300 adult females in Litania this year and finds the value of the sample mean.

1. State the probability of a Type I error. \includegraphics[max width=\textwidth, alt={}]{ff3433b0-baab-45e3-845e-56a794739bba-12_74_1577_557_322} ........................................................................................................................................ You should assume that the value of \(\sigma\) after 2004 remains at 6.95 .
2. Given that the value of \(\mu\) this year is actually 164.91 , find the probability of a Type II error.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

2 Jamie is conducting a hypothesis test on a random variable which has a normal distribution with standard deviation 1 The hypotheses are $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 5 \\ & \mathrm { H } _ { 1 } : \mu > 5 \end{aligned}$$ He takes a random sample of size 4
The mean of his sample is 6
He uses a 5\% level of significance.
Before Jamie conducted the test, what was the probability that he would make a Type I error? Circle your answer.
[0pt] [1 mark] \(0.0228 \quad 0.0456 \quad 0.0500 \quad 0.1587\)