Hypothesis Testing

Have a go first. Then open the working one line at a time.

1. Rank each variable

   Ranks of age: 2, 5, 3, 8, 6, 1, 7, 4 Ranks of price: 7, 3, 6, 1, 4, 8, 2, 5

   Rank 1 is the smallest value. Rank each list separately.

2. Correlate the ranks

   From the calculator: r_s = −0.976

   The calculator has a Spearman option. It correlates the ranks for you.

3. Interpret

   Strong negative correlation: older cars tend to cost less.

   r_s is close to −1, so the ranks are almost reversed. It is not exactly −1: two cars are out of order.

Have a go first. Then open the working one line at a time.

1. Average the tied ranks

   Ranks of maths: 2.5, 4.5, 2.5, 7, 1, 6, 4.5, 8 Ranks of music: 3, 5, 2, 5, 1, 8, 7, 5

   Two 56s would take ranks 2 and 3, so each gets 2.5. Three 45s take ranks 4, 5 and 6, so each gets 5.

2. Correlate the ranks

   From the calculator: r_s = 0.704

   Enter the ranks as two lists and find the correlation, or use the Spearman option.

3. Interpret

   Moderate to strong positive correlation: students who do well in maths tend to do well in music.

   Say 'tend to'. A correlation does not mean one mark causes the other.

Have a go first. Then open the working one line at a time.

1. Rank for each dancer

   Judge 1 ranks A to H: 2, 4, 1, 6, 7, 3, 8, 5 Judge 2 ranks A to H: 1, 3, 2, 5, 8, 4, 7, 6

   Find each dancer in each list. Dancer C is 1st for Judge 1 (rank 1) and 2nd for Judge 2 (rank 2).

2. Correlate the ranks

   From the calculator: r_s = 0.905

   Put the two rank lists in the calculator, in the same dancer order.

3. Interpret

   The judges largely agree.

   A value near 1 means the judges placed the dancers in nearly the same order.

Have a go first. Then open the working one line at a time.

1. Hypotheses

   H0: there is no correlation between hours revised and mark. H1: there is a positive correlation between hours revised and mark.

   'Positive correlation' makes the test one-tailed.

2. Calculate r_s

   Ranks of hours (the two 5s share 5.5): 3, 5.5, 2, 9, 7, 4, 8, 1, 5.5 From the calculator: r_s = 0.979

   The tied hours share an average rank.

3. Compare

   r_s = 0.979 > 0.600, so reject H0.

   For a positive test, reject H0 if r_s is greater than the critical value.

4. In context

   There is sufficient evidence, at the 5% significance level, of a positive correlation between hours revised and mark.

   Write the conclusion about hours and marks.

Have a go first. Then open the working one line at a time.

1. Hypotheses

   H0: there is no correlation between the numbers taking Chemistry and the numbers taking Art. H1: there is a negative correlation between the numbers taking Chemistry and the numbers taking Art.

   'Negative correlation' makes the test one-tailed, in the negative direction.

2. Calculate r_s

   Ranks of Chemistry: 5, 2.5, 7, 1, 6, 2.5, 4 Ranks of Art: 2, 6, 1, 7, 3, 4.5, 4.5 From the calculator: r_s = −0.936

   Both lists have ties, so use average ranks.

3. Compare

   r_s = −0.936 < −0.571, so reject H0.

   For a negative test, reject H0 if r_s is less than minus the critical value.

4. In context

   There is sufficient evidence, at the 10% significance level, of a negative correlation between the numbers taking Chemistry and the numbers taking Art.

   Schools with more Chemistry students tend to have fewer Art students.

Have a go first. Then open the working one line at a time.

1. Hypotheses

   H0: there is no correlation between homework hours and social media time. H1: there is a correlation between homework hours and social media time.

   'Any correlation' means the test is two-tailed: H1 allows positive or negative.

2. Calculate r_s

   From the calculator: r_s = −0.238

   Rank both lists first if you are entering ranks.

3. Compare

   |r_s| = 0.238 < 0.738, so do not reject H0.

   In a two-tailed test, compare the size of r_s, ignoring the sign, with the critical value.

4. In context

   There is insufficient evidence, at the 5% significance level, of a correlation between homework hours and social media time.

   Do not say there is no correlation. Say there is insufficient evidence of one.

Have a go first. Then open the working one line at a time.

1. Hypotheses

   H0: there is no correlation between the examiners' marks. H1: there is a positive correlation between the examiners' marks.

   Agreeing means that a high mark from one goes with a high mark from the other: positive correlation, one-tailed.

2. Calculate r_s

   Ranks of Examiner 1: 4, 7, 2, 6, 1, 3, 5 Ranks of Examiner 2: 4.5, 6, 2, 7, 3, 1, 4.5 From the calculator: r_s = 0.811

   The marks 15 and 15 for Examiner 2 are tied.

3. Compare

   r_s = 0.811 > 0.714, so reject H0.

   A positive one-tailed test.

4. In context

   There is sufficient evidence, at the 5% significance level, that the two examiners tend to agree.

   Agreement means a positive correlation between the two sets of marks.

Have a go first. Then open the working one line at a time.

1. Pearson's r

   r increases, closer to 1.

   The point (16, 150) was far above the rest. Once it is moved onto the pattern, the points lie closer to a straight line.

2. Spearman's r_s

   r_s stays the same: 0.964.

   The point was the largest y before and is still the largest y after (48 is more than 41). No rank changes, so r_s does not change. This is why r_s is less affected by outliers.

3. Changing units

   r_s does not change.

   Dividing every y by 1000 keeps the order of the values, so the ranks are the same.