Hypothesis test for a Normal mean

The cholesterol level, X, of an adult can be modelled by a normal distribution with mean 5.7 mmol/l and standard deviation 1.2 mmol/l.

The lower quartile of X is a and the upper quartile is b

It is given that the P(X ≤ a) = 0.25 and P(X ≥ b) = 0.25

Find the interquartile range of X

A group of scientists believe that a new dietary supplement can reduce the mean cholesterol level.

A random sample of 160 adults were given the new dietary supplement for a trial period of one year.

After the trial, their cholesterol level was found to have a mean of 5.6 mmol/l.

Carry out a hypothesis test at the 5% significance level to investigate whether the mean cholesterol level has reduced after taking the dietary supplement.

It can be assumed that the cholesterol level after taking the dietary supplement follows a normal distribution with an unchanged variance.

The mass of aluminium cans recycled each day in a city may be modelled by a normal distribution with mean 24 500 kg and standard deviation 5 200 kg.

State the probability that the mass of aluminium cans recycled on any given day is not equal to 24 500 kg.

To reduce costs, the city’s council decides to collect aluminium cans for recycling less frequently.

Following the decision, it was found that over a 24-day period a total mass of 641 520 kg of aluminium cans was recycled.

It can be assumed that the distribution of the mass of aluminium cans recycled is still normal with standard deviation 5 200 kg, and that the 24-day period can be regarded as a random sample.

Investigate, at the 5% level of significance, whether the mean daily mass of aluminium cans recycled has changed.

A member of the council claims that if a different sample of 24 days had been used the hypothesis test in part (b) would have given the same result.

Comment on the validity of this claim.

The number of working hours per week of employees in a company is modelled by a normal distribution with mean of 34 hours and a standard deviation of 4.5 hours.

The manager claims that the mean working hours per week of the company’s employees has increased.

A random sample of 30 employees in the company was found to have mean working hours per week of 36.2 hours.

Carry out a hypothesis test at the 2.5% significance level to investigate the manager’s claim.

A team game involves solving puzzles to escape from a room.

Using data from the past, the mean time to solve the puzzles and escape from one of these rooms is 65 minutes with a standard deviation of 11.3 minutes.

After recent changes to the puzzles in the room, it is claimed that the mean time to solve the puzzles and escape has changed.

To test this claim, a random sample of 100 teams is selected.

The total time to solve the puzzles and escape for the 100 teams is 6780 minutes.

Assuming that the times are normally distributed, test at the 2% level the claim that the mean time has changed.

It is known that a hospital has a mean waiting time of 4 hours for its Accident and Emergency (A&E) patients.

After some new initiatives were introduced, a random sample of 12 patients from the hospital’s A&E Department had the following waiting times, in hours.

Carry out a hypothesis test at the 10% significance level to investigate whether the mean waiting time at this hospital’s A&E department has changed.

You may assume that the waiting times are normally distributed with standard deviation 0.8 hours.

The graph below shows the amount of salt, in grams, purchased per person per week in England between 2001–02 and 2014, based upon the Large Data Set.

Meera and Gemma are arguing about what this graph shows.

Meera believes that the amount of salt consumed by people decreased greatly during this period.

Gemma says that this is not the case.

Using your knowledge of the Large Data Set, give two reasons why Gemma may be correct.

It is known that the mean amount of sugar purchased per person in England in 2014 was 78.9 grams, with a standard deviation of 25.0 grams.

In 2018, a sample of 918 people had a mean of 80.4 grams of sugar purchased per person.

Investigate, at the 5% level of significance, whether the mean amount of sugar purchased per person in England has changed between 2014 and 2018.

Assume that the survey data is a random sample taken from a normal distribution and that the standard deviation has remained the same.

Another test is performed to determine whether the mean amount of fat purchased per person has changed between 2014 and 2018.

At the 10% significance level, the null hypothesis is rejected.

With reference to the 10% significance level, explain why it is not necessarily true that there has been a change.

In a region of England, the government decides to use an advertising campaign to encourage people to eat more healthily.

Before the campaign, the mean consumption of chocolate per person per week was known to be 66.5 g, with a standard deviation of 21.2 g

After the campaign, the first 750 available people from this region were surveyed to find out their average consumption of chocolate.

State the sampling method used to collect the survey.

Explain why this sample should not be used to conduct a hypothesis test.

A second sample of 750 people revealed that the mean consumption of chocolate per person per week was 65.4 g

Investigate, at the 10% level of significance, whether the advertising campaign has decreased the mean consumption of chocolate per person per week.

Assume that an appropriate sampling method was used and that the consumption of chocolate is normally distributed with an unchanged standard deviation.