Large Data Set (Edexcel A Level Maths: Statistics): Exam Questions

Helen believes that the random variable , representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.

Write down the probability distribution for .

Using this model, find the probability that cloud cover is less than 50%.

Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than 50% was 0.315

Comment on the suitability of Helen’s model in the light of this information.

Suggest an appropriate refinement to Helen’s model.

A meteorologist is investigating the Daily Total Rainfall, mm, in Heathrow using a random sample of 120 days from the large data set.

The results are summarised in the table below.

On the grid below, draw a histogram to represent these data.

A "light-rain" day is defined as a day with rainfall between 1 mm and 5 mm.

Calculate an estimate for the number of "light-rain" days recorded in this sample.

Before producing the grouped frequency table, the meteorologist had to clean the data.

Using your knowledge of the large data set, explain why the daily total rainfall data needed to be cleaned.

Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below.

One of the 184 days is selected at random.

Find the probability that it has a daily mean total cloud cover of 6 or greater.

Magali is investigating whether the daily mean total cloud cover can be modelled using a binomial distribution.

She uses the random variable to denote the daily mean total cloud cover and believes that .

Using Magali’s model,

(i)  find

(ii)  find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7.

Explain whether or not your answers to part (b) support the use of Magali’s model.

There were 28 days that had a daily mean total cloud cover of 8.

For these 28 days, the daily mean total cloud cover for the following day is shown in the table below.

Find the proportion of these days when the daily mean total cloud cover was 6 or greater.

Comment on Magali’s model in light of your answer to part (d).

Ben is studying the Daily Total Rainfall, x mm, in Leeming for 1987.

He used all the data from the large data set and summarised the information in the following table.

Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation.

Using all 184 of these values, Ben estimates and

Calculate estimates for

(i) the mean Daily Total Rainfall,

(ii) the standard deviation of the Daily Total Rainfall.

Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987.

Using your knowledge of the large data set,

(i) give a reason why these data would not be suitable,

(ii) state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987.

The table below shows data from the large data set on the daily mean pressure, (hPa), and daily total sunshine, (hrs), in Camborne for a random sample of 12 days in 2015.

The equation of the regression line of on is .

Give an interpretation of the value of the gradient of the regression line.

Explain why it would not be reliable to use this regression equation to predict:

(i) the daily total sunshine on a day with a mean daily pressure of 980 hPa

(ii) the mean daily pressure on a day with 5.6 hours of total sunshine.

Summary statistics from the large data set for the daily mean windspeed (knots) measured in Heathrow throughout October 1987 and October 2015 are given in the table below.

Calculate the mean of the daily mean windspeeds for each of the two years.

The standard deviation for 2015 was 1.84.

Calculate the standard deviation for 1987 and compare the daily mean windspeeds for each of the two years.

The box plot in Figure 1 shows the Daily Mean Wind Speed, knots, for the 31 days in October 2015 in Hurn from the large data set.

Show that the value 13 is an outlier.

The Daily Mean Wind Speed data for Leuchars for the same period (October 2015) is summarised below.

Compare the Daily Mean Wind Speed in Hurn and Leuchars for October 2015.

A meteorologist wants to calculate the mean wind speed for Leuchars. The data in the large data set contains some entries recorded as "n/a".

State what "n/a" represents in the large data set and how the meteorologist should handle these entries.

The table below shows the daily total sunshine, hours, and the daily mean total cloud cover, oktas, for the first 10 days in May 2015 at Heathrow, taken from the large data set.

(i) Explain what is meant by 5 oktas of cloud cover.

(ii) Show that , given that there were 4 hours and 24 minutes of sunshine recorded at Heathrow on the first day of May 2015.

Calculate the product moment correlation coefficient, , between and .

(i) State suitable null and alternative hypotheses to test whether there is evidence of negative correlation between daily total sunshine and daily mean cloud cover.

(ii) Using the table of critical values for correlation coefficients in your formula booklet, find the critical value for this test at the 0.5% level of significance.

(iii) Test, at the 0.5% level of significance, whether there is evidence of a negative correlation between daily total sunshine and daily mean cloud cover at Heathrow in May 2015.

Stav is studying the large data set for September 2015.

He codes the variable Daily Mean Pressure, , using the formula .

The data for all 30 days from Hurn are summarised by

State the units of the variable .

Find the mean Daily Mean Pressure for these 30 days.

Find the standard deviation of Daily Mean Pressure for these 30 days.

Stav knows that, in the UK, winds circulate

• in a clockwise direction around a region of high pressure

• in an anticlockwise direction around a region of low pressure

The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015

The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order,

W     NE     E

You may assume that these 3 locations were under a single region of pressure.

Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table.

Give a reason for your answer.

Dian uses the large data set to investigate the Daily Total Rainfall, mm, for Camborne.

Write down how a value of is recorded in the large data set.

Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics

Use these statistics to calculate

(i) the mean of the Daily Total Rainfall in Camborne for August 2015,

(ii) the standard deviation of the Daily Total Rainfall in Camborne for August 2015.

Dian believes that the mean Daily Total Rainfall in August is less in the South of the UK than in the North of the UK.

The mean Daily Total Rainfall in Leuchars for August 2015 is 1.72 mm to 2 decimal places.

State, giving a reason, whether this provides evidence to support Dian's belief.

Dian uses the large data set to estimate the proportion of days with no rain in Camborne for 1987 to be 0.27 to 2 decimal places.

Explain why the distribution might not be a reasonable model for the number of days without rain for a 14‐day summer event.

Helen is studying one of the qualitative variables from the large data set for Heathrow from 2015.

She started with the data from 3rd May and then took every 10th reading.

There were only 3 different outcomes with the following frequencies

State the sampling technique Helen used.

From your knowledge of the large data set

(i) suggest which variable was being studied,

(ii) state the name of outcome .

George is also studying the same variable from the large data set for Heathrow from 2015.

He started with the data from 5th May and then took every 10th reading and obtained the following

Helen and George decided they should examine all of the data for this variable for Heathrow from 2015 and obtained the following

State what inference Helen and George could reliably make from their original samples about the outcomes of this variable at Heathrow, for the period covered by the large data set in 2015.

A random sample of 15 days is taken from the large data set for Perth in June and July 1987.

The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.

Describe the correlation.

The variable on the -axis is Daily Mean Temperature measured in °C.

Using your knowledge of the large data set,

(i) suggest which variable is on the -axis,

(ii) state the units that are used in the large data set for this variable.

Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow.

He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains .

Carry out a suitable test to investigate Stav’s belief at a 5% level of significance.

State clearly

• your hypotheses

• your critical value

On a random day at Heathrow the Daily Maximum Relative Humidity was 97%.

Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.

The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015.

An outlier is defined as a value

• more than below or

• more than above

The three lowest air temperatures in the data set are 7.6 °C, 8.1 °C and 9.1 °C.

The highest air temperature in the data set is 32.5 °C.

Complete the box plot in Figure 1 showing clearly any outliers.

Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come.

Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, °C, for Beijing in 2015

Show that, to 3 significant figures, the standard deviation is 5.19 °C.

Simon decides to model the air temperatures with the random variable

Using Simon’s model, calculate the 10th to 90th interpercentile range.

Simon wants to model another variable from the large data set for Beijing using a normal distribution.

State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer.

The large data set provides weather data for 184 consecutive days in each of the years 1987 and 2015. 

Describe how Charlie could take a systematic sample of 12 days from the data for Hurn for 1987 so that each date has a chance of being selected.

Charlie also takes a sample of 12 dates from the data for Hurn for 2015.

Using your knowledge of the large data set, explain why Charlie’s sample may not necessarily give him 12 numerical values to compare for each year.

Wendy is using the large data set to learn about the daily mean wind speeds for Leeming in June 2015. She lists the data below.

Using your knowledge of the large data set, state the units for the values in the table.

An outlier is defined as a value more than below or more than above .

On the grid below, draw a box plot for the information above.

Nguyen is using the large data set to investigate claims by a local newspaper that the chance of rain on a given day in Leuchars is 18%. She believes that the probability of it raining on a given day in Leuchars is different to 18%.

Nguyen includes all 368 days available in the large data set and finds that it rained on 80 of those days.

Using a 5% level of significance, test Nguyen's belief.  State your null and alternative hypotheses clearly.

Using your knowledge of the large data set, give two reasons why the sample may not be appropriate to test Nguyen's belief.

Medhi is a meteorologist investigating the weather in Heathrow and claims that there is negative correlation between the daily total rainfall, mm, and daily total sunshine, hours.

Medhi decides to use the large data set to investigate this claim and forms a sample using all the days in June 2015 relating to Heathrow.

Some values for the daily total rainfall, mm, are labelled as 'tr'. Using your knowledge of the large data set, state the range of values Medhi could assign to these values.

Medhi uses all the days in June 2015 as a sample and calculates the product moment correlation coefficient to be .

Carry out a suitable test at the 5% level of significance to investigate Medhi's claim. You should:

• state your hypotheses clearly

• state the critical value used

Medhi uses this data to calculate the equation for the regression line of on . He plans to use the regression line to estimate the amount of rainfall there will be in Heathrow during a day in December.

Give two reasons why this is unlikely to produce a reliable estimate.

David and Bowey are planning a trip in June to Beijing, Jacksonville or Perth.  The temperature of the city and the atmospheric pressure will be deciding factors, so they investigate these three cities using all of the data for June 2015 from the large data set.

Using all of the days in June 2015, the following summary statistics for the daily mean air temperatures ( ℃) and the daily mean pressure ( hPa) are calculated:

David also has the following information for Beijing in June:

  and 

Calculate the values of   and  .

David suffers from headaches when the atmospheric pressure changes quickly so he would like to choose a city where the pressure does not vary a lot. Additionally, Bowey does not like it when the temperature is higher than °C.

It is known that all the temperatures for Beijing in June 2015 were within standard deviations of the mean, whereas in Jacksonville there were temperatures that were higher than the mean by more than standard deviations.

(i) Use the data to explain why Bowey would not be happy visiting Jacksonville.

(ii) Hence, suggest with reasons which other city both David and Bowey would be happy to visit.

An ice cream shop owner in Camborne is trying to use data from the large data set alongside their own past sales data to help them estimate future sales. The mean daily temperature per month, °C, is shown with the mean daily number of ice creams sold per month, , from 2015 in the table below.

The equation for the regression line of on is .

Find an estimate for the expected total number of ice creams sold in the month of July if the average daily temperature for that month is 14.9 °C.

Suggest one other variable from the large data set which could be used to improve this model.

The ice cream shop owner claims that there is a causal link between and , and so if the shop sells more ice cream, the month will be hotter. 

Comment on this claim.

Roger has been looking at some data on the daily mean air temperature, t, in two different locations, Perth and Jacksonville, taken from the large data set.  All the data is taken from the month of July in 2015.

Unfortunately, some of the information has been lost and Roger does not know which data is for which location.

Complete the table.

Using your knowledge of the large data set, state which of the locations is most likely to be Jacksonville, giving a reason for your answer.

The table below shows the daily maximum relative humidity, rounded to the nearest per cent, for Leuchars between June and August .

Using your knowledge of the large data set, explain why roughly of these days contained fog and/or mist.

The data from the table are to be presented on a statistical diagram.

For a histogram, the frequency density for the – class is .

Find the frequency density for the – class.

For a cumulative frequency graph, state the coordinates of all the points that should be plotted.

Explain why an exact box plot cannot be drawn using only the information from the table.

Ororo, a meteorologist, is investigating the great storm of 1987 which largely affected the south of England. Ororo would like to compare the daily maximum gust in Hurn during the months of October 1987 and October 2015.

Using your knowledge of the large data set

(i) suggest one other city from the large data set that Ororo could use to investigate the great storm of 1987

(ii) state the units that are used in large data set to measure the daily maximum gust.

Ororo calculates the following summary statistics for the daily maximum gust in Hurn using the available data for October 1987.

An outlier is defined as a value which is more than 2 standard deviations away from the mean.

(i) Show that the maximum value in 1987 is an outlier.

(ii) Give a reason why Ororo should include the outlier when comparing the data from the two years.

Ororo calculates the following statistics for the daily maximum gust in Hurn for October 2015:

• Mean is 18.9 knots

• Standard deviation is 4.45 knows

Compare the daily maximum gust in Hurn for October 1987 and October 2015.