Hard A-Level Regression and correlation Questions

Challenging, exam-style A-Level Regression and correlation questions with worked solutions. Stretch yourself on the hardest pmcc, raw-data, exponential-model, prediction problems.

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A-Level34 questionsStep-by-step solutions
Question 1
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).
Question 2
8 markschallenging
A researcher designs a correlation study. A researcher wants to test whether there is a non-zero correlation using a sample of size n=16n = 16 with r=0.660r = 0.660. 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\neq 0

    A test for zero correlation.

  3. State the sample size and test type

    n=16, two-tailedn = 16,\ \text{two-tailed}

    Note the sample size and the tail.

  4. Write down the sample correlation coefficient

    r=0.660r = 0.660

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.4973\text{critical value} = 0.4973

    Read the value for this n and significance level.

  6. State the decision rule

    reject H0 if ∣r∣>0.4973\text{reject }H_0\text{ if } |r| > 0.4973

    Compare the statistic with the critical value.

  7. Form the test comparison

    0.6600>0.49730.6600 > 0.4973

    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

    two-tailed\text{two-tailed}

    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=16n = 16

    Larger samples give smaller critical values.

  13. Restate the observed statistic

    r=0.660r = 0.660

    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\neq 0

    These are the correct hypotheses for this test.

Answer
H0: ρ=0H_0:\ \rho = 0 and H1: ρ≠0H_1:\ \rho \neq 0 (a two-tailed test).
Question 3
8 markschallenging
A demographer studies two figures. A test of H0: ρ=0H_0:\ \rho = 0 against H1: ρ>0H_1:\ \rho > 0 uses a sample of size n=40n = 40, giving r=0.310r = 0.310. The critical value is 0.31200.3120. What is the correct conclusion?
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=40, one-tailed (positive)n = 40,\ \text{one-tailed (positive)}

    Note the sample size and the tail.

  4. Write down the sample correlation coefficient

    r=0.310r = 0.310

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.3120\text{critical value} = 0.3120

    Read the value for this n and significance level.

  6. State the decision rule

    reject H0 if r>0.3120\text{reject }H_0\text{ if } r > 0.3120

    Compare the statistic with the critical value.

  7. Form the test comparison

    0.3100≤0.31200.3100 \le 0.3120

    Place the statistic against the critical value.

  8. Decide whether it lies in the critical region

    not in the critical region\text{not in the critical region}

    This determines the outcome of the test.

  9. Make the statistical decision

    do not reject H0\text{do not reject } H_0

    State the decision on the null hypothesis.

  10. Consider the tail of the test

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

    The tail affects the critical value used.

  11. Relate the decision to the evidence

    no evidence of correlation\text{no evidence of correlation}

    Interpret the decision statistically.

  12. Note the role of the sample size

    n=40n = 40

    Larger samples give smaller critical values.

  13. Restate the observed statistic

    r=0.310r = 0.310

    Confirm the value used in the comparison.

  14. Summarise the outcome

    retain H0\text{retain } H_0

    This is the conclusion of the test.

  15. State the conclusion in context

    0.3100≤0.3120⇒do not reject H00.3100 \le 0.3120 \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
8 markschallenging
A chemist studies two concentrations. 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=−0.770r = -0.770. The critical value is 0.70790.7079. What is the correct conclusion?
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\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=−0.770r = -0.770

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.7079\text{critical value} = 0.7079

    Read the value for this n and significance level.

  6. State the decision rule

    reject H0 if ∣r∣>0.7079\text{reject }H_0\text{ if } |r| > 0.7079

    Compare the statistic with the critical value.

  7. Form the test comparison

    0.7700>0.70790.7700 > 0.7079

    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

    two-tailed\text{two-tailed}

    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=12n = 12

    Larger samples give smaller critical values.

  13. Restate the observed statistic

    r=−0.770r = -0.770

    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 conclusion in context

    0.7700>0.7079⇒reject H00.7700 > 0.7079 \Rightarrow \text{reject } H_0

    Report the decision about the correlation.

Answer
There is sufficient evidence to reject H0H_0; the data support a negative (non-zero) correlation.
Question 5
8 markschallenging
A financier studies two series. A test of H0: ρ=0H_0:\ \rho = 0 against H1: ρ≠0H_1:\ \rho \neq 0 uses a sample of size n=18n = 18, giving r=0.440r = 0.440. The critical value is 0.46830.4683. What is the correct conclusion?
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\neq 0

    A test for zero correlation.

  3. State the sample size and test type

    n=18, two-tailedn = 18,\ \text{two-tailed}

    Note the sample size and the tail.

  4. Write down the sample correlation coefficient

    r=0.440r = 0.440

    This is the observed statistic.

  5. Write down the critical value from tables

    critical value=0.4683\text{critical value} = 0.4683

    Read the value for this n and significance level.

  6. State the decision rule

    reject H0 if ∣r∣>0.4683\text{reject }H_0\text{ if } |r| > 0.4683

    Compare the statistic with the critical value.

  7. Form the test comparison

    0.4400≤0.46830.4400 \le 0.4683

    Place the statistic against the critical value.

  8. Decide whether it lies in the critical region

    not in the critical region\text{not in the critical region}

    This determines the outcome of the test.

  9. Make the statistical decision

    do not reject H0\text{do not reject } H_0

    State the decision on the null hypothesis.

  10. Consider the tail of the test

    two-tailed\text{two-tailed}

    The tail affects the critical value used.

  11. Relate the decision to the evidence

    no evidence of correlation\text{no evidence of correlation}

    Interpret the decision statistically.

  12. Note the role of the sample size

    n=18n = 18

    Larger samples give smaller critical values.

  13. Restate the observed statistic

    r=0.440r = 0.440

    Confirm the value used in the comparison.

  14. Summarise the outcome

    retain H0\text{retain } H_0

    This is the conclusion of the test.

  15. State the conclusion in context

    0.4400≤0.4683⇒do not reject H00.4400 \le 0.4683 \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.

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