The proportion of European men who are red-green colour-blind is $8\%$. How large a sample would need to be selected to be $95\%$ certain that it contains at least this proportion of red-green colour-blind men?

The masses of sweets produced by a machine are normally distributed with mean $\mu$ grams and standard deviation $1$ grams. A random sample of $65$ sweets produced by the machine has a mean mass of $29.6$ grams.

Find a $99\%$ confidence interval for $\mu$.

The masses of sweets produced by a machine are normally distributed with mean $\mu$ grams and standard deviation $1.0$ grams. A random sample of $65$ sweets produced by the machine has a mean mass of $29.6$ grams.

The manufacturer claims that the machine produces sweets with a mean mass of $30$ grams.

(ii) Use the confidence interval found in finding a $99\%$ confidence interval for $\mu$ to draw a conclusion about this claim.

Another random sample of $65$ sweets produced by the machine is taken. This sample gives a $99\%$ confidence interval that leads to a different conclusion from that found in part ii. Assuming that the value of $\mu$ has not changed, explain how this can be possible.

An employer who is being sued for the wrongful dismissal of an employee is advised that any award paid out will be based on national average earnings for employees of a similar age. A random sample of $120$ people is found to have a mean income of $21000$ with standard error $710$.

Find a $95\%$ confidence interval for the award.

A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of $15$ burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows.

$\begin{array}{lllllllllllllll}9& 7& 8& 9& 6& 11& 7& 9& 8& 9& 8& 10& 7& 9& 9\end{array}$

Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean $\mu$ and that the population standard deviation is $1.3$.

The manufacturer claims that the mean mass of fat in burgers of this type is $8 \mathrm{g}$. Use. $99\%$ confidence interval for $\mu$ to comment on this claim

A random sample of $n$ people were questioned about their internet use. $87$ of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is $0.1129<p<0.1771$.

Write down the mid-point of this confidence interval and hence find the value of $n$.

In order to obtain a random sample of people who live in her town, Jane chooses people at random from the telephone directory for her town.

Jane now uses a valid method to choose a random sample of $200$ people from her town and finds that $38$ live in apartments.

Calculate an approximate $99\%$ confidence interval for the proportion of all people in Jane's town who live in apartments.
 

In a survey a random sample of $150$ households in Nantville were asked to fill in a questionnaire about household budgeting.

The results showed that $33$ households owned more than one car. Find an approximate $99\%$ confidence interval for the proportion of all households in Nantville with more than one car.

Represent the union of two sets by Venn diagram for each of the following.

$X=\{x | x$ is a prime number between $80$ and $100\}$

$Y=\{y | y$ is an odd number between $90$ and $100\}$

A random sample of $n$ people were questioned about their internet use. $87$ of them had a high-speed internet connection. A confidence interval for the population proportion having a high-speed internet connection is $0.1129<p<0.1771$.

This interval is an $\alpha \%$ confidence interval. Find $\alpha$.

An employer who is being sued for the wrongful dismissal of an employee is advised that any award paid out will be based on national average earnings for employees of a similar age. A random sample of $120$ people is found to have a mean income of $21000$ with standard error $710$.

The employer wants to know the upper limit of the award that is very unlikely to be exceeded. The employer defines 'unlikely' as a probability of $0.001$.

Work out the unlikely upper limit of the award, giving your answer to the nearest dollar.

A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of $15$ burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows.

$\begin{array}{lllllllllllllll}9& 7& 8& 9& 6& 11& 7& 9& 8& 9& 8& 10& 7& 9& 9\end{array}$

Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean $\mu$ and that the population standard deviation is $1.3$.

Explain whether it was necessary to use the Central Limit Theorem in the calculation in calculating $99\%$ confidence interval for $\mu$.

Last year Samir found that the time for his journey to work had mean $45.7$ minutes and standard deviation $3.2$ minutes. Samir wishes to test whether his average journey time has increased this year. He notes the times, in minutes, for a random sample of $8$ journeys this year with the following results.

$\begin{array}{llllllll}46.2& 41.7& 49.2& 47.1& 47.2& 48.4& 53.7& 45.5\end{array}$

It may be assumed that the population of this year's journey times is normally distributed with standard deviation $3.2$ minutes.

State, with a reason, whether Samir should use a one-tail or a two-tail test.

In order to obtain a random sample of people who live in her town, Jane chooses people at random from the telephone directory for her town.

Jane now uses a valid method to choose a random sample of $200$ people from her town and finds that $38$ live in apartments.

Jane uses the same sample to give a confidence interval of width $0.1$ for this proportion. This interval is an $x\%$ confidence interval. Find the value of $x$.

The masses of sweets produced by a machine are normally distributed with mean $\mu$ grams and standard deviation $1.0$ grams. A random sample of $65$ sweets produced by the machine has a mean mass of $29.6$ grams.

The manufacturer claims that the machine produces sweets with a mean mass of $30$ grams.

Use the confidence interval found in finding a $99\%$ confidence interval for $\mu$ to draw a conclusion about this claim.

The volume, $v \mathrm{ml}$, of liquid dispensed by a vending machine for a random sample of $60$ hot drinks is summarised as follows:

$\sum v=17280$ and $\sum v^2=5015000$

Work out a $90\%$ confidence interval for the population mean.

Tennis balls of a certain brand are known to have a mean height of bounce of $64.7 \mathrm{cm}$, when dropped from a height of $100 \mathrm{cm}$. A change is made in the manufacturing process and it is required to test whether this change has affected the mean height of bounce. $100$ new tennis balls are tested and it is found that their mean height of bounce when dropped from a height of $100 \mathrm{cm}$ is $65.7 \mathrm{cm}$ and the unbiased estimate of the population variance is $15{ \mathrm{cm}}^2$.

Calculate a $95\%$ confidence interval for the population mean.

An employer who is being sued for the wrongful dismissal of an employee is advised that any award paid out will be based on national average earnings for employees of a similar age. A random sample of $120$ people is found to have a mean income of $21000$ with standard error $710$.

The employer wants to know the upper limit of the award that is very unlikely to be exceeded. The employer defines 'unlikely' as a probability of $0.001$.

Explain why the required size of the confidence interval is $99.8\%$.

Tennis balls of a certain brand are known to have a mean height of bounce of $64.7 \mathrm{cm}$, when dropped from a height of $100 \mathrm{cm}$. A change is made in the manufacturing process and it is required to test whether this change has affected the mean height of bounce. $100$ new tennis balls are tested and it is found that their mean height of bounce when dropped from a height of $100 \mathrm{cm}$ is $65.7 \mathrm{cm}$ and the unbiased estimate of the population variance is $15{ \mathrm{cm}}^2$.

Calculate a $95\%$ confidence interval for the population mean.

Explain what conclusion can be drawn about whether the change has affected the mean height of bounce.

A doctor wishes to investigate the mean fat content in low-fat burgers. He takes a random sample of $15$ burgers and sends them to a laboratory where the mass, in grams, of fat in each burger is determined. The results are as follows.

$\begin{array}{lllllllllllllll}9& 7& 8& 9& 6& 11& 7& 9& 8& 9& 8& 10& 7& 9& 9\end{array}$

Assume that the mass, in grams, of fat in low-fat burgers is normally distributed with mean $\mu$ and that the population standard deviation is $1.3$.

Calculate a $99\%$ confidence interval for $\mu$.