13D InClass

A man using his phone while driving

Credit: iStock/PeopleImages

The study in this in-class activity uses a single group of participants and measures their reaction times (in milliseconds) while performing a driving task without using their cell phones and then again while using their cell phones.[1]

We will use statistical inference to compare reaction times for the two conditions (using and not using cell phones).

 

 

Question 1

1) Do you think a driver’s reaction time is different when they are using a cell phone as  opposed to when they are not using a cell phone while driving?

The data collected include the reaction times (in milliseconds) with and without the use  of cell phones for 32 randomly selected participants. The data for the first 10  participants are displayed in the following table.

Participant Reaction Time (in milliseconds)
Cell Phone Use = Yes Cell Phone Use = No
1 636 604
2 623 556
3 615 540
4 672 522
5 601 459
6 600 544
7 542 513
8 554 470
9 543 556
10 520 531

Question 2

2) In order to use the study data to compare reaction times for the two conditions  (using and not using cell phones) using statistical inference, we must determine if  the two samples are dependent (paired) or independent samples. Analyze the  context of the study and determine if the samples are dependent or independent  samples. Explain.

In order to compare the reaction times for dependent samples, we first calculate the difference between the reaction time of each participant with the use of a cell phone and  without the use of a cell phone.

The reaction times of a participant performing the same driving task with and without the use of a cell phone in a randomly decided order are recorded in the following table. The  difference is defined by:

Difference = (reaction time with the use of a cell phone) – (reaction time without the use  of a cell phone)

Question 3

3) Calculate the missing difference for Participant 1 between the reaction times  performing the same driving task with and without the use of a cell phone.

Participant Reaction Time (in Milliseconds) Difference
Cell Phone Use = Yes Cell Phone Use = No
1 636 604
2 623 556 67
3 615 540 75
4 672 522 150
5 601 459 142
6 600 544 56
7 542 513 29
8 554 470 84
9 543 556 -13
10 520 531 −11
11 609 599 10
12 559 537 22
13 595 619 −24
14 565 536 29
15 573 554 19
16 554 467 87
17 626 525 101
18 501 508 −7
19 574 529 45
20 468 470 −2
21 578 512 66
22 560 487 73
23 525 515 10
24 647 499 148
25 456 448 8
26 688 558 130
27 679 589 90
28 960 814 146
29 558 519 39
30 482 462 20
31 527 521 6
32 536 543 −7

Question 4

4) Describe the difference, in context, for Participant 32.

Question 5

5) Look at the reaction times and the difference for Participant 8. Complete the  following sentence by circling the correct word in parentheses.

The reaction time for Participant 8 performing the same driving task was 84 milliseconds (shorter/longer) with the use of a cell phone than without the use of a  cell phone.

Question 6

6) Go to the DCMP Describing and Exploring Quantitative Variables tool at  https://dcmathpathways.shinyapps.io/EDA_quantitative/. Locate the “Enter Data” drop-down menu and select “Your Own.” Copy and paste the numbers in the  Difference column of the previous table.

Write down the sample mean difference and the sample standard deviation of the  differences.

A dependent (paired) t-test compares the mean difference to a hypothesized value,  which is often 0 (no difference). It is always important to check the assumptions of a test  before you perform any calculations.

Question 7

7) Given the previous information, is a dependent t-test the appropriate method to use  to answer the following research question: “Does a driver’s reaction times (in  milliseconds) differ when they are using a cell phone as opposed to when they are  not using a cell phone?”

Question 8

8) Go to the DCMP Compare Two Means tool at https://dcmathpathways.shinyapps.io/2sample_mean/ and click on the tab Two  Dependent Samples. Use the following inputs:

  • In the “Dataset” drop-down menu, choose “Reaction Times (Paired Experiment).”
  • In the left column, go to the drop-down menu for “Type of Inference” and select  “Significance Test.”

a) How do the sample mean difference and the sample standard deviation  difference you recorded in Question 6 compare to the descriptive statistics  calculated by the data analysis tool?

b) Complete the following table using the null and alternative hypotheses for  this research question.

Hypothesis Notation
Null There is no difference between the mean  reaction times while using a cell phone and  while not using a cell phone. [latex]H_{0}[/latex]
Alternative [latex]H_{A}[/latex]

The test statistic for the dependent (paired) t-test is calculated using the following formulas:

[latex]standard\;error\;of\;the\;difference=\frac{s_{d}}{\sqrt{n}}[/latex]

[latex]test\;statistic\;(t)=\frac{estimator-null\;value}{standard\;error\;of\;estimator}=\frac{\bar{d}-null\;value}{standard\;error\;of\;difference}[/latex]

Question 9

9) Using the output from the DCMP Compare Two Means tool:

a) What is the standard error of the estimated difference and the test statistic?  Verify the values using the previous formulas.

b) What is the P-value?

c) Write a conclusion in the given context.

Question 10

10) Go back to the DCMP Compare Two Means tool and change the “Type of  Inference” to “Confidence Interval.”

a) Calculate and interpret the 95% confidence interval.

b) What does the confidence interval tell you about a driver’s reaction time with  and without a cell phone? Write a tweet-length (240 characters or less)  public service announcement that uses this result.

c) Explain the connection between the results of the dependent t-test and the  95% confidence interval.

  1. Strayer, D. L. & Johnston, W. A. (2001, November 1). Driven to distraction: Dual-task studies of  simulated driving and conversing on a cellular telephone. Psychological Science, 12(6), 462–466. DOI:  10.1111/1467-9280.00386 ↵