Stats Medic - Sampling Distribution of Slopes Answer Key_ Does seat location matter? Part 1

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Does seat location matter? Part 1

Do students who sit in the front rows do better than students who sit farther away? A teacher randomly assigned 30 students to seats at the beginning of the semester and then recorded their exam scores at the end of the semester. Here are the results: Row 1 1 Score 76 77

1 94

1 1 1 2 2 2 2 2 2 3 3 3 3 3 3 99 88 90 83 85 74 79 77 79 90 88 68 78 83 79

Row 4 4 4 4 4 4 5 5 5 5 5 5 Score 94 72 101 70 63 76 76 65 67 96 79 96 1. Is this an observational study or an experiment? Why?

Experiment

,

because a treatment (row assigned) is being

imposed

.

2. Why is it important to randomly assign students to seats rather than letting them choose their own seat?

It allows us to show causation

.

2 Are they categorical or quantitative?_____________ 3. How many variables are we measuring?_____ quantitative Row score What is the explanatory variable (x)?__________________ Response variable(y)?_____________ 4. Use www.stapplet.com or your calculator to make a scatterplot. Sketch it below.

y

a +

fy.int] -

^

by [

slope

ROW

SCORE

85.95-1.5170×88

5. Find the least squares regression line (LSRL):__________________________________________ ya@ =

1.517 6. What is the slope of the LSRL:_________ -

Interpret the slope in the context of the problem.

For each additional row , a student 's scout goes doer

by 1.517 points

predicted

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Does the negative slope provide convincing evidence that sitting in a lower row causes higher exam scores, or is it plausible that the association is purely by chance because of random assignment?

In order to answer this question, we need to know more about “purely by chance because of random assignment”. If we assume that seat location has NO effect on exam score, then we could just randomly assign all 30 exam scores to each of the seat locations. We will do this by writing down each of the 30 exam scores onto an index card, shuffle the index cards, and then randomly assign them to seat locations. to:B O B true slope of population LSRL Null Shuffle up the note cards and randomly assign 6 students into each of the 5 rows. Record the results: - --

hypothesis

Answers vary

-

→

Row 1: _____, _____, _____, _____, _____, _____ Row 2: _____, _____, _____, _____, _____, _____ Row 3: _____, _____, _____, _____, _____, _____ Row 4: _____, _____, _____, _____, _____, _____ Row 5: _____, _____, _____, _____, _____, _____ Now find the slope of the LSRL:____________ Repeat this process 2 more times for a total of 3 different random assignments. Record the results.

Row 1: _____, _____, _____, _____, _____, _____ Row 2: _____, _____, _____, _____, _____, _____ Row 3: _____, _____, _____, _____, _____, _____ Row 4: _____, _____, _____, _____, _____, _____ Row 5: _____, _____, _____, _____, _____, _____ Now find the slope of the LSRL:____________ Row 1: _____, _____, _____, _____, _____, _____ Row 2: _____, _____, _____, _____, _____, _____ Row 3: _____, _____, _____, _____, _____, _____ Row 4: _____, _____, _____, _____, _____, _____ Row 5: _____, _____, _____, _____, _____, _____ Now find the slope of the LSRL:____________

Use Two Quantitative variables applet at www.stapplet.com

You have now calculated three different possible values for the slope based on random assignment.

Take these 3 values to the dotplot on the whiteboard in the front of the room. When everyone in class has recorded their data, copy the dotplot below:

Randomization Distribution of b -

P value = -

Because

Yodo

o

Tse - n N ( O , Ob )

=

0.156

0.156 > 0.05

SEb to estimate Ob

-

fail to reject Ho

.

1

result Our This study source was downloaded by 100000853497421 from CourseHero.com on 04-27-2023 14:59:15 GMT -05:00 bi 1.517 -

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Sampling Distribution of b Important ideas:

guying

left

LT

statisticparameter y intercept

a

d

slope

b

B

-

Higgs

tired

"

.

%

sampling Distribution of b

Q.se

of

.

.

.

Check Your Understanding You may have heard that your nose and ears grow through your whole life. While it is true that your nose and ears get bigger throughout life, its not because they grow, but because of gravity. The cartilage in your nose and ears break down as we age and the “growth” people observe is the result of drooping. To quantify the expansion of ears over time, a random sample of 30 adults were selected. For each adult, their age (in years) was recorded and their ear height (cm) was measured.

Below is the regression output. Is there evidence of a positive linear relationship between age and ear height? Assume the conditions for inference are met. x

'

y

=

atbx

y

TEA

are

X

b-

→ ←

JEB

SD of residuals

a. What is the estimate for α? Interpret this value. a = 2.8871

y intercept -

years old (newborns) we height to be 2.8871cm on average

for people that are predict their ear

0

,

b. What is the estimate for β? Interpret this value.

b- 0.0021

slope

of 1 year in age we predict height to increase by 0.0021cm

For each increase

the

ear

,

.

c. What is the estimate for σ? Interpret this value.

5=0.3613

The actual ear heights

SD of residuals

0.3613

cm

typically vary by

from the predicted ear

d. Give the standard error of the slope SEb. Interpret this value.

SEB -0.0059 The slope of the sample from the slope of the about 0.0059

LSRL

cmlyear

.

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height .

typically varies

population

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about

LSRL

by

.