Recover sample stats from CI

3 Based on a random sample of 700 people living in a certain area, a confidence interval for the proportion, \(p\), of all people living in that area who had travelled abroad was found to be \(0.5672 < p < 0.6528\).

1. Find the proportion of people in the sample who had travelled abroad.
2. Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.

2 The lengths of a random sample of 50 roads in a certain region were measured.Using the results,a \(95 \%\) confidence interval for the mean length,in metres,of all roads in this region was found to be[245,263].

1. Find the mean length of the 50 roads in the sample.
2. Calculate an estimate of the standard deviation of the lengths of roads in this region.
3. It is now given that the lengths of roads in this region are normally distributed.
   State,with a reason,whether this fact would make any difference to your calculation in part(b).

3 A survey of a random sample of \(n\) people found that 61 of them read The Reporter newspaper. A symmetric confidence interval for the true population proportion, \(p\), who read The Reporter is \(0.1993 < p < 0.2887\).

1. Find the mid-point of this confidence interval and use this to find the value of \(n\).
2. Find the confidence level of this confidence interval.

2 The weights in grams of oranges grown in a certain area are normally distributed with mean \(\mu\) and standard deviation \(\sigma\). A random sample of 50 of these oranges was taken, and a \(97 \%\) confidence interval for \(\mu\) based on this sample was (222.1, 232.1).

1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
2. Estimate the sample size that would be required in order for a \(97 \%\) confidence interval for \(\mu\) to have width 8 .

2 Shane is studying the lengths of the tails of male red kangaroos. He takes a random sample of 14 male red kangaroos and measures the length of the tail, \(x \mathrm {~m}\), for each kangaroo. He then calculates a \(90 \%\) confidence interval for the population mean tail length, \(\mu \mathrm { m }\), of male red kangaroos. He assumes that the tail lengths are normally distributed and finds that \(1.11 \leqslant \mu \leqslant 1.14\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

2 A rowing club has a large number of members.A random sample of 12 of these members is taken and the pulse rate,\(x\) beats per minute(bpm),of each is measured after a 30 -minute training session.A \(98 \%\) confidence interval for the population mean pulse rate,\(\mu \mathrm { bpm }\) ,is calculated from the sample as \(64.22 < \mu < 68.66\) .

1. Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) .
2. State an assumption that is necessary for the confidence interval to be valid. \includegraphics[max width=\textwidth, alt={}, center]{b5ff998a-fcb6-4a1b-ae86-ec66b0dccc3c-04_2718_38_141_2009}

6 Nassa is researching the lengths of a particular type of snake in two countries, \(A\) and \(B\).

1. He takes a random sample of 10 snakes of this type from country \(A\) and measures the length, \(x \mathrm {~m}\), of each snake. He then calculates a \(90 \%\) confidence interval for the population mean length, \(\mu \mathrm { m }\), for snakes of this type, assuming that snake lengths have a normal distribution. This confidence interval is \(3.36 \leqslant \mu \leqslant 4.22\). Find the sample mean and an unbiased estimate for the population variance.
2. Nassa also measures the lengths, \(y \mathrm {~m}\), of a random sample of 8 snakes of this type taken from country \(B\). His results are summarised as follows. $$\sum y = 27.86 \quad \sum y ^ { 2 } = 98.02$$ Nassa claims that the mean length of snakes of this type in country \(B\) is less than the mean length of snakes of this type in country \(A\). Nassa assumes that his sample from country \(B\) also comes from a normal distribution, with the same variance as the distribution from country \(A\). Test at the \(10 \%\) significance level whether there is evidence to support Nassa's claim.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1 A basketball club has a large number of players. The heights, \(x \mathrm {~m}\), of a random sample of 10 of these players are measured. A \(90 \%\) confidence interval for the population mean height, \(\mu \mathrm { m }\), of players in this club is calculated. It is assumed that heights are normally distributed. The confidence interval is \(1.78 \leqslant \mu \leqslant 2.02\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample.

7 A random sample of 8 sunflower plants is taken from the large number grown by a gardener, and the heights of the plants are measured. A 95\% confidence interval for the population mean, \(\mu\) metres, is calculated from the sample data as \(1.17 < \mu < 2.03\). Given that the height of a sunflower plant is denoted by \(x\) metres, find the values of \(\Sigma x\) and \(\Sigma x ^ { 2 }\) for this sample of 8 plants.

Petra is studying a particular species of bird. She takes a random sample of 12 birds from nature reserve \(A\) and measures the wing span, \(x \mathrm {~cm}\), for each bird. She then calculates a \(95 \%\) confidence interval for the population mean wing span, \(\mu \mathrm { cm }\), for birds of this species, assuming that wing spans are normally distributed. Later, she is not able to find the summary of the results for the sample, but she knows that the \(95 \%\) confidence interval is \(25.17 \leqslant \mu \leqslant 26.83\). Find the values of \(\sum x\) and \(\sum x ^ { 2 }\) for this sample. Petra also measures the wing spans of a random sample of 7 birds from nature reserve \(B\). Their wing spans, \(y \mathrm {~cm}\), are as follows. $$\begin{array} { l l l l l l l } 23.2 & 22.4 & 27.6 & 25.3 & 28.4 & 26.5 & 23.6 \end{array}$$ She believes that the mean wing span of birds found in nature reserve \(A\) is greater than the mean wing span of birds found in nature reserve \(B\). Assuming that this second sample also comes from a normal distribution, with variance the same as the first distribution, test, at the \(10 \%\) significance level, whether there is evidence to support Petra's belief.

7 The pulse rate of each member of a random sample of 25 adult UK males who exercise for a given period each week is measured in beats per minute. A \(98 \%\) confidence interval for the mean pulse rate, \(\mu\) beats per minute, for all such UK males was calculated as \(61.21 < \mu < 64.39\), based on a \(t\)-distribution.

1. Calculate the sample mean pulse rate and the standard deviation used in the calculation.
2. State an assumption necessary for the validity of the confidence interval.
3. The mean pulse rate for all UK males is 72 beats per minute. State, giving a reason, if it can be concluded that, on average, UK males who exercise have a reduced pulse rate.

A random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95% confidence interval for the population mean height, \(\mu\) metres, is calculated from the data as \(1.65 \leqslant \mu \leqslant 1.85\).

1. Find an unbiased estimate for the population variance. [3]
2. Denoting the height of a member of the club by \(x\) metres, find \(\Sigma x^2\) for this sample of 9 members. [4]

A normally distributed variable, \(X\), has unknown mean \(\mu\) and unknown standard deviation \(\sigma\). A sample of 10 values of \(X\) was taken. From these 10 values, a 95% confidence interval for \(\mu\) was calculated to be $$(30.47, 32.93)$$ Use this confidence interval to find unbiased estimates for \(\mu\) and \(\sigma^2\). [4 marks]