CHS Stats - Chapter 9

correlation

relationship between 2 variables

independent variable

aka explanatory variable; x

dependent variable

aka response variable; y

types of correlation

correlation coefficient

a measure of the strength and direction of a line of a linear relationship between variables

• r = sample correlation coefficient

• p = population correlation coefficient (Greek letter Rho - “row”)

correlation coefficient direction

based on +/- r

correlation coefficient strength

based on |r|; the closer it is to 1, the stronger it is; if |r| =

• 0.00-0.24 = no correlation

• 0.25-0.49 = weak correlation

• 0.50-0.74 = moderate correlation

• 0.76-1.00 = strong correlation

• 1.00 = perfect correlation

Hypothesis Test for Correlation Steps/Formulas

1. Ho: p = 0 - no significant linear correlation; Ha: p ≠ 0 - there is a significant linear correlation; specify claim

2. α =

3. Standardized Test Statistic: t = r / √[(1-r²) / (n-2)]

4. P-Value: P = tcdf(lower, upper, df) * 2

   • df = n - 2

   • * 2 bc it’s always 2-tailed

5. Decision: If P ≤ α, reject Ho; if P > α, fail to reject Ho

6. Statement about the claim: can change it up, but always use evidence

   • Ex. R Ho… evidence suggests that there is a relationship between age and vocab

Is reject Ho a relationship or no relationship? What so fail to reject Ho?

• Reject Ho: relationship

• Fail to reject Ho: no relationship/no correlation

regression line

aka line of best fit, the line for which the sum of the squares of the residuals is a minimum

predicted value

every point on a regression line

residual

d sub i; the difference between an observed y and a predicted y; d sub i = observed y - predicted y

• when the predicted value is on the line, the residual = 0

(x̄, ȳ)

the point that includes the mean of all the x-coordinates and the mean of all the y-coordinates; is always on the regression line

ŷ

predicted y-value based on x sub 0; begins the regression line equation (ŷ = ax+b)

What do you say for predictions for x-values that are too far outside the range of x-values in the given data?

Not meaningful

explained variation

variation that can be explained by the nature of the relationship between x and y

unexplained variation

variation that is not explained by the x-y relationship. Instead, it’s due to other factors, like sampling error, coincidence, and lurking/confounding variables.

total variation

the explained variation + unexplained variation

coefficient of determination

r²; the ratio of explained variation to total variation

• r² = explained variation/total variation

Explanation statement for coefficient of determination

r² as a percent of the variation in y can be explained by the relationship between x and y.

• substitute problem-specific values/variables for the underlined portions

Prediction Interval for ŷ

1. Input x & y values into L1 and L2, respectively

2. Find the regression line

3. Find the predicted ŷ value based on the given x sub 0 value

4. Find s sub e (Stat - Tests - F: LinRegTTest, use the s), n, ∑x, and ∑x² (in 1-Var-Stats - just L1 for List, leave FreqList blank)

5. Find E (equation given on test)

   • t sub c = invT(area to the left, df)

     • df = n - 2

       • n = # of data pairs

6. (ŷ - E, ŷ + E)

s sub e

Standard Error of the Estimate