Explanatory variable: independent or x variable
Response variable: dependent or y variable
Form: linear, non-linear, or no relationship
Direction: positive or negative
Strength: strong, moderate, weak
Perfect ± relationship: ±1
Strong ± relationship: ±0.75 to ±1; it can be concluded that y inc/dec as x increases
Moderate ± relationship: ±0.5 to ±0.75; there is some evidence to suggest that y inc/dec as x increases
Weak ± relationship: ±0.25 to ±0.5; there is limited evidence to suggest that y inc/dec as x increases
No relationship: –0.25 to 0.25
Correlation coefficient (r) determines direction and strength and is only reliable if data is linear with no outliers.
The coefficient of determination gives the percentage change in the RV that can be explained by the DV. The higher
the r2, the stronger the relationship between x and y, and the better the regression line fits.
ex. if r = -0.42, r2 = (-0.42)2; 17% of change in y can be explained by change in x, and 83% is unexplained.
Least squared regression line: y = a + bx, e.g. height = 100 + 2.5 x age. This can be used to make predictions; ex.
what is the predicted height for an eight-year-old? 100 + 2.5 x 8 = 120
ex. interpret the slope for the equation p = 0.5 + – 31.4w. for every 1 unit increase in w (RV), p (EV) increases by 0
Interpolation: prediction made within the original range of the data given; more reliable. Extrapolation = vice versa
Residuals: the difference between the actual value and the predicted value. Positive residual: predicted value is below
the actual result; an underestimate and vice versa. ex. calculate the residual of a 12 year old who is 142 cm tall. 100 +
2.5 x 12 = 130 → 142 – 130 = 12; positive residual as it is above the line
Comment on the appropriateness of fitting a linear model. If there is no clear pattern in the residual, it is appropriate.
Comment on the validity of the prediction. State whether it is interpolation or extrapolation.
X has a residual of ±2.6. what information does this provide about the EV? X is 2.6% below/above predicted EV
Arithmetic sequences
Rule for d: d = t2 − t1 = t3 − t2 = t4 − t3 = …, OR d = tn − tn−1
Rule for nth term: tn = a + (n-1)d
Recurrence relation: tn+1 = tn + d, where t1 = a
ex. if the first 6 terms of a sequence are 2, 7, 12, 17, 22, 27, state the value of the 10 th and 20th terms. tn = a + (n-1)d
t10 = 2 + (10-1)5 = 47 and t20 = 2 + (20-1)5 = 97
ex. for the arithmetic sequence 54, 51, 48, …, find which term equals zero. tn = a + (n-1)d → tn = 54 + (n-1)-3 →
0 = 54 + (n-1)-3 → 54 ÷ 3 = n-1 → 18 + 1 = n → n = 19
ex. the temperature at a ski resort was displayed as 3°C, but steadily decreased by 0.6°C every hour. how long after
opening is the temperature at 0°C? tn = a + (n-1)d → 0 = 3 + (n-1)-0.6 → 3 ÷ 0.6 = n-1 → n = 6 hours
ex. find a rule for the nth term of sequence tn+1 = tn - 8, where t1 = 4. tn = a + (n-1)d → tn = 4 – 8 (n-1) → tn = 12 – 9n
ex. find the arithmetic sequence in which t10 = 14 and t25 = 105. tn = a + (n-1)d → 14 = a + 9d, 105 = a + 22d →
105 = 14 – 9d + 22d → 105 – 14 = -9d + 22d → → d = 7 → a = 14 – 9(7) → a = 49, ∴ tn+1 = tn + 7, where t1 = 49
Geometric sequences
Rule for r: r = t2 / t1 = t3 / t2 = t4 / t3 = …, OR r = tn / tn-1
Rule for nth term: tn = arn-1
Recurrence relation: tn+1 = rtn, where t1 = a
ex. if the first 6 terms of a geometric sequence are 2, 6, 18, 243, 729, 2187, state the value of the 8 th and 15th terms.
tn = arn-1 → t8 = 2(3)8-1 = 4374, t15 = 2(3)15-1 = 9565938
ex. a population increases by 11% each month. if population in the first month was 20, what will there be in 2 years?
11% = 100 + 11 = 1.11 → tn = arn-1 → t24 = 20(1.11)24-1 = 220.525
ex. in 1 year, $25000 is raised for a club. in 2 years, $20000 is raised, and $16000 in 3. How much is raised in 5?
tn = arn-1 → t5 = 25000(0.8)5-1 → t5 = $10240
ex. state the geometric sequence tn = 4(0.25)n-1, where n = 1, 2, 3, as a recurrence relation. tn+1 = 0.25tn, where t1 = 16
ex. the fourth term in a geometric sequence is 128 and the common ration is two; find the first term. t n = arn-1
128 = a(2)4-1 →128 ÷ 8 = a(8) → a = 16, ∴ tn+1 = 2tn, where t1 = 16
Recurrence relations
First order: tn+1 = rtn + d
ex. find the first 6 terms defined by the difference equation tn+1 = tn – 4, where t1 = 12. t2 = t1 – 4, ∴ 12 – 4 = 8 →
t3 = t2 – 4, ∴ 8 – 4 = 4 → t4 = t3 – 4, ∴ 4 – 4 = 0 → t5 = t4 – 4, ∴ 0 – 4 = -4 → t6 = t5 – 4, ∴ -4 – 4 = -8
Response variable: dependent or y variable
Form: linear, non-linear, or no relationship
Direction: positive or negative
Strength: strong, moderate, weak
Perfect ± relationship: ±1
Strong ± relationship: ±0.75 to ±1; it can be concluded that y inc/dec as x increases
Moderate ± relationship: ±0.5 to ±0.75; there is some evidence to suggest that y inc/dec as x increases
Weak ± relationship: ±0.25 to ±0.5; there is limited evidence to suggest that y inc/dec as x increases
No relationship: –0.25 to 0.25
Correlation coefficient (r) determines direction and strength and is only reliable if data is linear with no outliers.
The coefficient of determination gives the percentage change in the RV that can be explained by the DV. The higher
the r2, the stronger the relationship between x and y, and the better the regression line fits.
ex. if r = -0.42, r2 = (-0.42)2; 17% of change in y can be explained by change in x, and 83% is unexplained.
Least squared regression line: y = a + bx, e.g. height = 100 + 2.5 x age. This can be used to make predictions; ex.
what is the predicted height for an eight-year-old? 100 + 2.5 x 8 = 120
ex. interpret the slope for the equation p = 0.5 + – 31.4w. for every 1 unit increase in w (RV), p (EV) increases by 0
Interpolation: prediction made within the original range of the data given; more reliable. Extrapolation = vice versa
Residuals: the difference between the actual value and the predicted value. Positive residual: predicted value is below
the actual result; an underestimate and vice versa. ex. calculate the residual of a 12 year old who is 142 cm tall. 100 +
2.5 x 12 = 130 → 142 – 130 = 12; positive residual as it is above the line
Comment on the appropriateness of fitting a linear model. If there is no clear pattern in the residual, it is appropriate.
Comment on the validity of the prediction. State whether it is interpolation or extrapolation.
X has a residual of ±2.6. what information does this provide about the EV? X is 2.6% below/above predicted EV
Arithmetic sequences
Rule for d: d = t2 − t1 = t3 − t2 = t4 − t3 = …, OR d = tn − tn−1
Rule for nth term: tn = a + (n-1)d
Recurrence relation: tn+1 = tn + d, where t1 = a
ex. if the first 6 terms of a sequence are 2, 7, 12, 17, 22, 27, state the value of the 10 th and 20th terms. tn = a + (n-1)d
t10 = 2 + (10-1)5 = 47 and t20 = 2 + (20-1)5 = 97
ex. for the arithmetic sequence 54, 51, 48, …, find which term equals zero. tn = a + (n-1)d → tn = 54 + (n-1)-3 →
0 = 54 + (n-1)-3 → 54 ÷ 3 = n-1 → 18 + 1 = n → n = 19
ex. the temperature at a ski resort was displayed as 3°C, but steadily decreased by 0.6°C every hour. how long after
opening is the temperature at 0°C? tn = a + (n-1)d → 0 = 3 + (n-1)-0.6 → 3 ÷ 0.6 = n-1 → n = 6 hours
ex. find a rule for the nth term of sequence tn+1 = tn - 8, where t1 = 4. tn = a + (n-1)d → tn = 4 – 8 (n-1) → tn = 12 – 9n
ex. find the arithmetic sequence in which t10 = 14 and t25 = 105. tn = a + (n-1)d → 14 = a + 9d, 105 = a + 22d →
105 = 14 – 9d + 22d → 105 – 14 = -9d + 22d → → d = 7 → a = 14 – 9(7) → a = 49, ∴ tn+1 = tn + 7, where t1 = 49
Geometric sequences
Rule for r: r = t2 / t1 = t3 / t2 = t4 / t3 = …, OR r = tn / tn-1
Rule for nth term: tn = arn-1
Recurrence relation: tn+1 = rtn, where t1 = a
ex. if the first 6 terms of a geometric sequence are 2, 6, 18, 243, 729, 2187, state the value of the 8 th and 15th terms.
tn = arn-1 → t8 = 2(3)8-1 = 4374, t15 = 2(3)15-1 = 9565938
ex. a population increases by 11% each month. if population in the first month was 20, what will there be in 2 years?
11% = 100 + 11 = 1.11 → tn = arn-1 → t24 = 20(1.11)24-1 = 220.525
ex. in 1 year, $25000 is raised for a club. in 2 years, $20000 is raised, and $16000 in 3. How much is raised in 5?
tn = arn-1 → t5 = 25000(0.8)5-1 → t5 = $10240
ex. state the geometric sequence tn = 4(0.25)n-1, where n = 1, 2, 3, as a recurrence relation. tn+1 = 0.25tn, where t1 = 16
ex. the fourth term in a geometric sequence is 128 and the common ration is two; find the first term. t n = arn-1
128 = a(2)4-1 →128 ÷ 8 = a(8) → a = 16, ∴ tn+1 = 2tn, where t1 = 16
Recurrence relations
First order: tn+1 = rtn + d
ex. find the first 6 terms defined by the difference equation tn+1 = tn – 4, where t1 = 12. t2 = t1 – 4, ∴ 12 – 4 = 8 →
t3 = t2 – 4, ∴ 8 – 4 = 4 → t4 = t3 – 4, ∴ 4 – 4 = 0 → t5 = t4 – 4, ∴ 0 – 4 = -4 → t6 = t5 – 4, ∴ -4 – 4 = -8
