Exponential models reduced with logs, the product moment correlation coefficient, hypothesis tests for correlation with critical values, interpreting output.
12 original questions, 75 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.
Ruby records the mean daily screen time, s hours, and the mean nightly sleep, h hours, for a random sample of 12 Year 12 students. The product moment correlation coefficient between s and h is −0.614
State what is measured by the product moment correlation coefficient.
Test, at the 5% level of significance, whether there is evidence of negative correlation between screen time and sleep. State your hypotheses clearly and the critical value used.
4 marks
Show answer
a) the strength of the linear relationship (linear correlation) between two variablesb) H0: ρ = 0, H1: ρ < 0; critical value -0.4973; -0.614 < -0.4973 so reject H0: there is evidence of negative correlation between screen time and sleep
2
A head of sixth form believes there is a relationship between the number of hours a week, w, that Year 13 students spend in part-time work and their mock exam score, m. She takes a random sample of 20 Year 13 students and finds that the product moment correlation coefficient between w and m is −0.402
Test, at the 5% level of significance, whether there is evidence of correlation between w and m. State your hypotheses clearly and the critical value used.
4 marks
Show answer
H0: ρ = 0, H1: ρ ≠ 0; critical values ±0.4438; |-0.402| < 0.4438 so do not reject H0: insufficient evidence of correlation between part-time work and mock score
3
A microbiologist grows a culture of bacteria. She counts the number of bacteria, N, at time t hours, and models the data by N = abt, where a and b are constants.
She codes the data using y = log10N and finds the regression line of y on t to be
y = 2.15 + 0.087t
Find the value of a and the value of b, giving your answers to 3 significant figures.
Interpret the value of b in this context.
4 marks
Show answer
a) a = 141, b = 1.22b) the number of bacteria is multiplied by 1.22 each hour (it grows by about 22% an hour)
4
Laila uses records for a random sample of 40 days at a weather station in Wales. She finds that the product moment correlation coefficient between daily mean wind speed and daily mean temperature is −0.284
She carries out a test for negative correlation, and the p-value for her test is 0.0379
Write down suitable hypotheses for Laila’s test.
Using a 5% level of significance, state the conclusion of her test in context.
State, giving a reason, whether the conclusion would change if a 1% level of significance were used.
Laila says “This proves that strong winds cause lower temperatures.” Comment on her statement.
5 marks
Show answer
a) H0: ρ = 0, H1: ρ < 0, where ρ is the population PMCCb) 0.0379 < 0.05 so reject H0: there is evidence of negative correlation between wind speed and temperaturec) yes: 0.0379 > 0.01, so at 1% she would not reject H0 (insufficient evidence)d) correlation does not imply causation; the correlation is weak, and both may depend on another factor such as the weather system or the season
5
A sprint coach measures the reaction time, x milliseconds, and the 60 m sprint time, y seconds, for a random sample of 9 athletes.
x
182
195
201
210
214
226
233
241
250
y
7.42
7.61
7.40
7.66
7.52
7.58
7.82
7.55
7.80
Use your calculator to find the product moment correlation coefficient between x and y.
Test, at the 5% level of significance, whether there is evidence of positive correlation between reaction time and sprint time. State your hypotheses and the critical value used.
State, giving a reason, whether your conclusion would be different if the coach had tested for any correlation at the 5% level.
6 marks
Show answer
a) r = 0.675 (0.6747)b) H0: ρ = 0, H1: ρ > 0; critical value 0.5822; 0.6747 > 0.5822 so reject H0: evidence of positive correlationc) no: the two-tailed critical value is 0.6664 and 0.6747 > 0.6664, so H0 is still rejected
6
Ayo is studying the mass, m kg, and the resting heart rate, h beats per minute, of 14 species of mammal. He codes the data using
x = log10my = log10h
The product moment correlation coefficient between x and y is −0.962
Test, at the 1% level of significance, whether there is evidence of negative correlation between x and y. State your hypotheses clearly and the critical value used.
The equation of the regression line of y on x is y = 2.36 − 0.24x
Find a model for h in terms of m in the form h = amk, where a and k are constants to be found.
Use your model to estimate the resting heart rate of a mammal of mass 70 kg.
7 marks
Show answer
a) H0: ρ = 0, H1: ρ < 0; critical value -0.6120; -0.962 < -0.6120 so reject H0: evidence of negative correlation between x and yb) h = 229m(-0.24)c) about 83 beats per minute (82.6)
7
A start-up company launches a new revision app. The number of downloads, D, in week w was recorded for the first 10 weeks. The company models D by
D = kcw
where k and c are constants. The regression line of log10D on w is
log10D = 1.82 + 0.135w
Find the values of k and c, giving your answers to 3 significant figures.
Interpret the value of c in context.
Use the regression line to find the first week in which the model predicts more than 2000 downloads.
Comment on the reliability of your answer to part (c).
7 marks
Show answer
a) k = 66.1, c = 1.36b) the number of downloads is multiplied by about 1.36 each week (an increase of about 36% a week)c) week 11d) week 11 is just outside the range of the data (weeks 1 to 10), so it is a small extrapolation; only reliable if the exponential growth continues, which may not happen
8
Priya records the altitude, a metres, and the mean July temperature, t °C, at a random sample of 10 weather stations in Scotland. The altitudes range from 20 m to 950 m. The product moment correlation coefficient between a and t is −0.931
Test, at the 1% level of significance, whether there is evidence that altitude and temperature are negatively correlated. State your hypotheses clearly and the critical value used.
State, with a reason, which variable is the explanatory variable.
The regression line of t on a is t = 15.8 − 0.0062a
Interpret the gradient of the regression line in context.
Estimate the mean July temperature at a weather station at an altitude of 600 m.
Explain why it would not be sensible to use the line to estimate the mean July temperature at the summit of Ben Nevis, 1345 m above sea level.
7 marks
Show answer
a) H0: ρ = 0, H1: ρ < 0; critical value -0.7155; -0.931 < -0.7155 so reject H0: evidence of negative correlationb) altitude a: altitude can affect temperature, not the other way roundc) for each extra metre of altitude, the mean July temperature falls by 0.0062 °C (about 0.62 °C per 100 m)d) 12.1 °Ce) 1345 m is outside the range of the data (20 m to 950 m), so it is extrapolation
9
A researcher is planning a study of the link between the number of steps walked per day and resting blood pressure. A pilot study suggests that the sample correlation coefficient will be about r = 0.42
Using the table of critical values, find the smallest sample size, n, for which r = 0.42 would give significant evidence of positive correlation at the 5% level.
In the full study, she takes a random sample of 25 adults and obtains r = 0.42
Test, at the 5% level of significance, whether there is evidence of correlation between the two variables. State your hypotheses clearly and the critical value used.
Explain why a significant result does not mean that the correlation is strong.
6 marks
Show answer
a) n = 17 (critical value 0.4124 0.42)b) H0: ρ = 0, H1: ρ ≠ 0; critical values ±0.3961; 0.42 > 0.3961 so reject H0: evidence of correlationc) with a large enough sample even a weak correlation is significant; r = 0.42 is only weak to moderate, and the test only shows that ρ is unlikely to be zero
10
A chemist measures the concentration, C mg per litre, of a substance t minutes after a reaction starts.
t
0
2
4
6
8
10
12
C
81.2
57.6
43.1
30.4
22.6
15.9
11.7
Use your calculator to find the product moment correlation coefficient between (i) t and C (ii) t and y, where y = log10C
Explain why your answers to part (a) suggest that C = abt is a suitable model.
Find the equation of the regression line of y on t, giving the constants to 4 decimal places.
Hence find the values of a and b to 3 significant figures.
Using your model, find the time taken for the concentration to halve.
8 marks
Show answer
a) (i) -0.964 (ii) -1.00 (-0.9998)b) r for t and log C is closer to -1, so log C is almost exactly linear in t, which is what C = abt givesc) y = 1.9076 - 0.0700t (gradient -0.07005)d) a = 80.8, b = 0.851e) 4.30 minutes
11
The value, £V, of a model of car is modelled by V = abt, where t is the age of the car in years and a and b are constants. The graph shows the straight line of log10V against t. The line passes through the points (0, 4.30) and (8, 3.62).
Find the value of a and the value of b, giving your answers to 3 significant figures.
Interpret the value of b in context.
Find, according to the model, the age at which the car’s value first falls below £5000
Give one reason why the model may not be suitable for very old cars.
7 marks
Show answer
a) a = 20 000 (19 952.6), b = 0.822b) the car loses about 17.8% of its value each year (its value is multiplied by 0.822 each year)c) 7.07 yearsd) e.g. the model predicts the value tends to 0, but an old car keeps some value (scrap value), or classic cars can gain value
12
Ellie records the number of pupils, P, at a school of 600 pupils who have caught a winter virus by the end of day d of an outbreak.
d
1
2
3
4
5
6
7
8
P
6
9
13
21
30
47
68
104
Ellie suggests the model P = abd, where a and b are constants.
Show that this model gives a straight line when log10P is plotted against d.
Find the product moment correlation coefficient between d and log10P, and test, at the 5% level of significance, whether there is evidence of positive correlation between them. State your hypotheses and the critical value used.
Find the regression line of log10P on d and hence find the values of a and b, to 3 significant figures.
Use the model to find the first day on which more than 500 pupils would have caught the virus, and comment on this prediction.
10 marks
Show answer
a) log P = log a + d log b, which is of the form y = c + mx with y = log P, x = db) r = 0.9997 (1.00); H0: ρ = 0, H1: ρ > 0; critical value 0.6215; 0.9997 > 0.6215 so reject H0: evidence of positive correlationc) log P = 0.5981 + 0.1773d; a = 3.96, b = 1.50d) day 12; this is extrapolation beyond day 8, and with only 600 pupils the growth must slow as fewer pupils are left to catch it, so the prediction is unreliable (likely too early)
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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.