A-Level Regression and correlation Practice Questions

Free A-Level Regression and correlation practice questions with full step-by-step worked solutions. Covers pmcc, summary-statistics, regression-line, prediction. Practise exam-style problems and check your method.

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A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
For a set of bivariate data, Sxx=40S_{xx} = 40, Syy=30S_{yy} = 30 and Sxy=32S_{xy} = 32. Calculate the product moment correlation coefficient rr, giving your answer to 3 decimal places.
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Worked solution

  1. Recall the formula for the PMCC

    r=SxySxxSyyr=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}

    The product moment correlation coefficient uses the three summary statistics.

  2. Write down the given summary statistics

    Sxx=40, Syy=30, Sxy=32S_{xx}=40,\ S_{yy}=30,\ S_{xy}=32

    These values are provided in the question.

  3. State the PMCC to 3 decimal places

    r=3240×30=0.924r=\frac{32}{\sqrt{40\times 30}}=0.924

    This is the required correlation coefficient.

Answer
0.9240.924
Question 2
2 markseasy
A regression model was built from data with xx between 1010 and 4040. The model is used to predict a value when x=25x = 25. Which statement is correct?
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Worked solution

  1. State the range of x in the data

    10≤x≤4010 \le x \le 40

    The data only cover this interval of x.

  2. State the value to predict at

    x=25x = 25

    This is where we want an estimate.

  3. State the conclusion

    interpolation: reliable\text{interpolation: reliable}

    This is the reliability judgement.

Answer
Predicting at x=25x = 25 is interpolation (inside the data range 1010 to 4040), so the estimate is reasonably reliable.
Question 3
3 marksintermediate
An economist studies two indices. A test of H0: ρ=0H_0:\ \rho = 0 against H1: ρ≠0H_1:\ \rho \neq 0 uses a sample of size n=12n = 12, giving r=0.400r = 0.400. The critical value is 0.57600.5760. What is the correct conclusion?
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Worked solution

  1. Define the population correlation coefficient

    ρ=population PMCC\rho = \text{population PMCC}

    Set up the parameter being tested.

  2. State the hypotheses

    H0: ρ=0,H1: ρ≠0H_0:\ \rho = 0,\quad H_1:\ \rho\neq 0

    A test for zero correlation.

  3. State the sample size and test type

    n=12, two-tailedn = 12,\ \text{two-tailed}

    Note the sample size and the tail.

  4. Write down the sample correlation coefficient

    r=0.400r = 0.400

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.5760\text{critical value} = 0.5760

    Read the value for this n and significance level.

  6. State the conclusion in context

    0.4000≤0.5760⇒do not reject H00.4000 \le 0.5760 \Rightarrow \text{do not reject } H_0

    Report the decision about the correlation.

Answer
There is insufficient evidence to reject H0H_0; the correlation could be zero.
Question 4
5 markshard
A trainer plans a study. A researcher wants to test whether there is a positive correlation using a sample of size n=14n = 14 with r=0.580r = 0.580. Which are the correct hypotheses?
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Worked solution

  1. Define the population correlation coefficient

    ρ=population PMCC\rho = \text{population PMCC}

    Set up the parameter being tested.

  2. State the hypotheses

    H0: ρ=0,H1: ρ>0H_0:\ \rho = 0,\quad H_1:\ \rho>0

    A test for zero correlation.

  3. State the sample size and test type

    n=14, one-tailed (positive)n = 14,\ \text{one-tailed (positive)}

    Note the sample size and the tail.

  4. Write down the sample correlation coefficient

    r=0.580r = 0.580

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.4575\text{critical value} = 0.4575

    Read the value for this n and significance level.

  6. State the decision rule

    reject H0 if r>0.4575\text{reject }H_0\text{ if } r > 0.4575

    Compare the statistic with the critical value.

  7. Form the test comparison

    0.5800>0.45750.5800 > 0.4575

    Place the statistic against the critical value.

  8. Decide whether it lies in the critical region

    in the critical region\text{in the critical region}

    This determines the outcome of the test.

  9. Make the statistical decision

    reject H0\text{reject } H_0

    State the decision on the null hypothesis.

  10. State the hypotheses to be used

    H0: ρ=0, H1: ρ>0H_0:\ \rho = 0,\ H_1:\ \rho>0

    These are the correct hypotheses for this test.

Answer
H0: ρ=0H_0:\ \rho = 0 and H1: ρ>0H_1:\ \rho > 0 (a one-tailed (positive) test).
Question 5
8 markschallenging
A clinician designs a correlation study. A researcher wants to test whether there is a negative correlation using a sample of size n=22n = 22 with r=−0.550r = -0.550. Which are the correct hypotheses?
Show worked solution

Worked solution

  1. Define the population correlation coefficient

    ρ=population PMCC\rho = \text{population PMCC}

    Set up the parameter being tested.

  2. State the hypotheses

    H0: ρ=0,H1: ρ<0H_0:\ \rho = 0,\quad H_1:\ \rho<0

    A test for zero correlation.

  3. State the sample size and test type

    n=22, one-tailed (negative)n = 22,\ \text{one-tailed (negative)}

    Note the sample size and the tail.

  4. Write down the sample correlation coefficient

    r=−0.550r = -0.550

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.4227\text{critical value} = 0.4227

    Read the value for this n and significance level.

  6. State the decision rule

    reject H0 if r<−0.4227\text{reject }H_0\text{ if } r < -0.4227

    Compare the statistic with the critical value.

  7. Form the test comparison

    −0.5500<−0.4227-0.5500 < -0.4227

    Place the statistic against the critical value.

  8. Decide whether it lies in the critical region

    in the critical region\text{in the critical region}

    This determines the outcome of the test.

  9. Make the statistical decision

    reject H0\text{reject } H_0

    State the decision on the null hypothesis.

  10. Consider the tail of the test

    one-tailed (negative)\text{one-tailed (negative)}

    The tail affects the critical value used.

  11. Relate the decision to the evidence

    evidence of correlation\text{evidence of correlation}

    Interpret the decision statistically.

  12. Note the role of the sample size

    n=22n = 22

    Larger samples give smaller critical values.

  13. Restate the observed statistic

    r=−0.550r = -0.550

    Confirm the value used in the comparison.

  14. Summarise the outcome

    reject H0\text{reject } H_0

    This is the conclusion of the test.

  15. State the hypotheses to be used

    H0: ρ=0, H1: ρ<0H_0:\ \rho = 0,\ H_1:\ \rho<0

    These are the correct hypotheses for this test.

Answer
H0: ρ=0H_0:\ \rho = 0 and H1: ρ<0H_1:\ \rho < 0 (a one-tailed (negative) test).

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