Scatter diagrams

Amar is studying the flight of a bird from its nest. He measures the bird’s height above the ground, h metres, at time t seconds for 10 values of t Amar finds the equation of the regression line for the data to be h = 38.6 – 1.28t

Interpret the gradient of this line. The product moment correlation coefficient between h and t is – 0.510

Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight. You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

Jane draws the following scatter diagram for Amar’s data.

With reference to the scatter diagram, state, giving a reason, whether or not the regression line h = 38.6 – 1.28 t is an appropriate model for these data. Jane suggests an improved model using the variable u = t k² where k is a constant. She obtains the equation h = 38.1 – 0.78u

Choose a suitable value for k to write Jane’s improved model for h in terms of t only.

Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person

• their resting heart rate, h beats per minute

• the number of minutes, m, spent exercising each week

Her results are shown on the scatter diagram.

Interpret the nature of the relationship between h and m Anna codes the data using the formulae

The product moment correlation coefficient between x and y is – 0.897

Test whether or not there is significant evidence of a negative correlation between x and y You should

• state your hypotheses clearly

• use a 5% level of significance

• state the critical value used

The equation of the line of best fit of y on x is y = – 0.05 x + 1.92

Use the equation of the line of best fit of y on x to find a model for h on m in the form

where a and k are constants to be found.

TOTAL FOR STATISTICS IS 50 MARKS

Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, x, in their last name

• the number of letters, y, in their first name

His results are shown in the scatter diagram on the next page.

Describe the correlation between x and y. Marc suggests that parents with long last names tend to give their children shorter first names.

Using the scatter diagram comment on Marc’s suggestion, giving a reason for your answer. The results from Marc’s random sample of 16 observations are given in the table below.

Use your calculator to find the product moment correlation coefficient between x and y for these data.

Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should

• state your hypotheses clearly

• use a 5% level of significance

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 x-axis is Daily Mean Temperature measured in °C.

Using your knowledge of the large data set,

suggest which variable is on the y-axis,

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 r = −0.377

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.

A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 different nests and tests for pollutant level, p, and measures the thinning of the shell, t. The results are shown in the table below.

[You may use p² = 1967 and pt = 694]

Draw a scatter diagram on the axes on page 7 to represent these data.

Explain why a linear regression model may be appropriate to describe the relationship between p and t.

Calculate the value of S_pt and the value of S_pp.

Find the equation of the regression line of t on p, giving your answer in the form t = a + bp.

Plot the point ( ) and draw the regression line on your scatter diagram.

The scientist reviews similar studies and finds that pollutant levels above 16 are likely to result in the death of a chick soon after hatching.

Estimate the minimum thinning of the shell that is likely to result in the death of a chick.

On a particular day the height above sea level, x metres, and the mid-day temperature, y °C, were recorded in 8 north European towns. These data are summarised below

Find S yy

Calculate, to 3 significant figures, the product moment correlation coefficient for these data.

Give an interpretation of your coefficient.

A student thought that the calculations would be simpler if the height above sea level, h, was measured in kilometres and used the variable h instead of x.

Write down the value of S hh

Write down the value of the correlation coefficient between h and y.

A travel agent sells flights to different destinations from Beerow airport. The distance d, measured in 100 km, of the destination from the airport and the fare £ f are recorded for a random sample of 6 destinations.

Using the axes below, complete a scatter diagram to illustrate this information.

Explain why a linear regression model may be appropriate to describe the relationship between f and d.

Calculate the equation of the regression line of f on d giving your answer in the form f a + bd.

Give an interpretation of the value of b.

Jane is planning her holiday and wishes to fly from Beerow airport to a destination km t away. A rival travel agent charges 5p per km.

Find the range of values of t for which the first travel agent is cheaper than the rival.