Find minimum sample size

4 A certain train journey takes place every day throughout the year. The time taken, in minutes, for the journey is normally distributed with variance 11.2.

1. The mean time for a random sample of \(n\) of these journeys was found. A \(94 \%\) confidence interval for the population mean time was calculated and was found to have a width of 1.4076 minutes, correct to 4 decimal places. Find the value of \(n\).
2. A passenger noted the times for 50 randomly chosen journeys in January, February and March. Give a reason why this sample is unsuitable for use in finding a confidence interval for the population mean time.
3. A researcher took 4 random samples and a \(94 \%\) confidence interval for the population mean was found from each sample. Find the probability that exactly 3 of these confidence intervals contain the true value of the population mean.

2 The heights of a certain species of animal have been found to have mean 65.2 cm and standard deviation 7.1 cm . A researcher suspects that animals of this species in a certain region are shorter on average than elsewhere. She takes a large random sample of \(n\) animals of this species from this region and finds that their mean height is 63.2 cm . She then carries out an appropriate hypothesis test.

1. She finds that the value of the test statistic \(z\) is - 2.182 , correct to 3 decimal places.
   1. Stating a necessary assumption, calculate the value of \(n\).
   2. Carry out the hypothesis test at the \(4 \%\) significance level.
   3. Explain why it was necessary to use the Central Limit theorem in carrying out the test.

1 The result of a memory test is known to be normally distributed with mean \(\mu\) and standard deviation 1.9. It is required to have a \(95 \%\) confidence interval for \(\mu\) with a total width of less than 2.0 . Find the least possible number of tests needed to achieve this.

2 Heights of a certain species of animal are known to be normally distributed with standard deviation 0.17 m . A conservationist wishes to obtain a \(99 \%\) confidence interval for the population mean, with total width less than 0.2 m . Find the smallest sample size required.

4 The distance a particular football player runs in a match is modelled by a normal distribution with standard deviation 0.3 kilometres. A random sample of \(n\) matches is taken.
The distance the player runs in this sample of matches has mean 10.8 kilometres.
The sample is used to construct a \(93 \%\) confidence interval for the mean, of width 0.0543 kilometres, correct to four decimal places. 4

1. Find the value of \(n\) 4
2. Find the \(93 \%\) confidence interval for the mean, giving the limits to three decimal places.
   4
3. Alison claims that the population mean distance the player runs is 10.7 kilometres. She carries out a hypothesis test at the 7\% level of significance using the random sample and the hypotheses $$\begin{aligned} & \mathrm { H } _ { 0 } : \mu = 10.7 \\ & \mathrm { H } _ { 1 } : \mu \neq 10.7 \end{aligned}$$ 4 (c) (i) State, with a reason, whether the null hypothesis will be accepted or rejected. 4 (c) (ii) Describe, in the context of the hypothesis test in part (c)(i), what is meant by a Type II error. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-06_2488_1730_219_141}

3 In a random sample of 100 voters from a constituency, 32 said that they would support the Cyan Party.

1. Find an approximate \(99 \%\) confidence interval for the proportion of voters in the constituency who would support the Cyan Party.
2. Using the given sample proportion, estimate the smallest size of sample needed for the width of a \(99 \%\) confidence interval to be less than 0.04 .

35% of a random sample of \(n\) students walk to college. This result is used to construct an approximate 98% confidence interval for the population proportion of students who walk to college. Given that the width of this confidence interval is 0.157, correct to 3 significant figures, find \(n\). [5]

A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are 205 \quad 310 \quad 405 \quad 195 \quad 320.

1. Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. [4]

It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.

1. Find the minimum sample size required. [6]

(Total 10 marks)

A company produces climbing ropes. The lengths of the climbing ropes are normally distributed. A random sample of 5 ropes is taken and the length, in metres, of each rope is measured. The results are given below. 120.3 \quad 120.1 \quad 120.4 \quad 120.2 \quad 119.9

1. Calculate unbiased estimates for the mean and the variance of the lengths of the climbing ropes produced by the company. [5]

The lengths of climbing rope are known to have a standard deviation of 0.2 m. The company wants to make sure that there is a probability of at least 0.90 that the estimate of the population mean, based on a random sample size of \(n\), lies within 0.05 m of its true value.

1. Find the minimum sample size required. [6]

A film-buff is interested in how long it takes for the credits to roll at the end of a movie. She takes a random sample of 20 movies from those that she has bought on DVD and finds that the credits on these films last for a total of 46 minutes and 15 seconds

1. Assuming that the time for the credits to roll follows a Normal distribution with a standard deviation of 23 seconds, use her data to calculate a 90% confidence interval for the mean time taken for the credits to roll. [5]
2. Find the minimum number of movies she would need to have included in her sample for her confidence interval to have a width of less than 10 seconds. [5]
3. Explain why her sample might not be representative of all movies. [1]