Large Data Set (Edexcel AS Maths: Statistics): Exam Questions

Jiang is studying the variable Daily Mean Pressure from the large data set.

He drew the following box and whisker plot for these data for one of the months for one location using a linear scale but

• he failed to label all the values on the scale

• he gave an incorrect value for the median

Using your knowledge of the large data set, suggest a suitable value for the median.

(You are not expected to have memorised values from the large data set. The question is simply looking for sensible answers.)

Using your knowledge of the large data set, suggest a suitable value for the range.

(You are not expected to have memorised values from the large data set. The question is simply looking for sensible answers.)

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.

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.

Fred and Nadine are investigating whether there is a linear relationship between Daily Mean Pressure, hPa, and Daily Mean Air Temperature, °C, in Beijing using the 2015 data from the large data set.

Fred randomly selects one month from the data set and draws the scatter diagram in Figure 1 using the data from that month.

The scale has been left off the horizontal axis.

Describe the correlation shown in Figure 1.

Nadine chooses to use all of the data for Beijing from 2015 and draws the scatter diagram in Figure 2.

She uses the same scales as Fred.

Explain, in context, what Nadine can infer about the relationship between and using the information shown in Figure 2.

Using your knowledge of the large data set, state a value of for which interpolation can be used with Figure 2 to predict a value of .

Using your knowledge of the large data set, explain why it is not meaningful to look for a linear relationship between Daily Mean Wind Speed (Beaufort Conversion) and Daily Mean Air Temperature in Beijing in 2015.

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.

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.

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).

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.

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.

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.

Szilard wants to compare the average daily mean temperature in Beijing in both 1987 and 2015 using the large data set.

Szilard selects the data from the first 10 days in October in both 1987 and 2015.

Give two reasons why this sampling method is not suitable for Szilard's investigation.

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.

A selection of data from the large data set relating to the mean daily air temperature in Beijing for the first 10 days in October in both 1987 and 2015 is given above.  Climate activists use temperature data to track changes over time.

Using the data given above, find the mean of the daily mean air temperature for both 1987 and 2015.

Give one reason why the sample used above should not be used to draw wider conclusions about how the temperature in China has changed from 1987 to 2015.

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.

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.

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.

A selection of data from the large data set relating to the daily mean cloud cover, measured in oktas, in Heathrow for the first 10 days in May 1987 is given below.

7         4         5         2         7         4          2         0          3          5

Using your knowledge of the large data set, explain why a value of 10 oktas would be impossible.

Find:

(i) the value of the median of the data,

(ii) the interquartile range of the data. 

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 incomplete box plot below shows data from the large data set regarding cloud cover between May and October 2015 in Cambourne.  Cloud cover is measured in Oktas on a scale from 0 (no cloud cover) to 8 (full cloud cover).

Find the interquartile range.

An outlier is defined as any data value that falls either more than 1.5  (interquartile range) above the upper quartile or less than 1.5  (interquartile range) below the lower quartile.

(i) Find the boundaries (fences) at which outliers are defined.

(ii) Explain why, using your knowledge of how cloud cover is measured in the large data set, there cannot be any high valued outliers.

Complete the box plot given that, where appropriate, the maximum and minimum values should be located at the boundaries (fences) at which outliers are defined. (You are not required to mark any outliers on the box plot.)

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.

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.

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.

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.