LEARN · EXPLAIN · REVISE

Explain whether adding t to all y-values changes r.

Read the idea, work independently, then explain what changed.

高三選擇性必修第三册(A版).pdf · 8.1 · PDF 98 / printed page 93

TOPIC 01

Correlation of paired data

Interpret scatter patterns and compute a correlation from centered paired summaries.

Interpret scatter patterns and compute a correlation from centered paired summaries.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

t=10

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the paired-data summaries, preserving which observations belong together.

Hint 2

Use this intermediate relation.

(y_i+t)-(\bar y+t)=y_i-\bar y

Worked solution

Use the paired-data summaries, preserving which observations belong together.
Apply the stated relation and retain its conditions.
$S_{xy},S_{yy}\text{ unchanged}$
Correlation is invariant under translation because centered deviations are unchanged.

The requested relation or conclusion is shown below.

r\text{ unchanged}

Checks and common pitfalls: Correlation is invariant under translation because centered deviations are unchanged.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Interpret scatter patterns and compute a correlation from centered paired summaries.
Which condition is essential in correlation of paired data?
Can a clear U-shaped relation have zero Pearson correlation?

Defining relation: Interpret scatter patterns and compute a correlation from centered paired summaries.
$r=S_{xy}/\sqrt{S_{xx}S_{yy}}$
Conditions: Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.