In a classic study, Loftus and Palmer (1974) investigated - Math

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In a classic study, Loftus and Palmer (1974) investigated the relationship between memory for eyewitnesses and the questions they are asked. In the study, participants watched a film of an automobile accident and then were questioned about the accident. One group was asked how fast the cars were going when they "smashed into" each other. A second group was asked about the speed when the cars "hit" each other, and a third group was not asked any question about the speed of the cars. A week later, the participants returned to answer additional questions about the accident, including whether they recalled seeing any broken glass. Although there was no broken glass in the film, several students claimed to remember seeing it. The following table shows the frequency distribution of responses for each group.

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13) a) We are investigating the relationship between memory for eye witness and the question they were asked.

We want to see if the proportion of participants who claim to remember broken glass differ significantly from group to group, ie we want to test

H_0: memory for eye witness and the question they were asked are independent.

vs H_1: they are not independent.

We will use the $\chi^2$ -test for independence.

The test statistic is

$\chi^2=\sum_{i=1}^{n}\frac{(O_i-E_i)^2}{E_i}$ where, O_i: observed frequency, E_i: expected frequency, no. of cells

The expected frequencies are calculated as

$E_i=\frac{R\times C}{Total}$ where, & are the $i^{th}$ cell's corresponding row & column total.

We prepare the following table:-

	Response About	Broken Glass
	Yes	No
Smashed Into
Hit
Control (Not asked)
Total

The observed value of the test statistic is

$\chi^2_{obs}=\sum_{i=1}^{6}\frac{(O_i-E_i)^2}{E_i}=7.778$

The critical value from the $\chi^2$ -table is $\chi^2_{0.05;\, (3-1)(2-1)}=\chi^2_{0.05;\, 2}=5.991$

As the observed value of the test statistic is greater than the critical value of $\chi^2_{0.05;2}$ , we reject H_0 at 5% level of significance and conclude that memory for eye witness and the question they were asked are dependent, ie the proportion of participants who claim to remember broken glass differ significantly from group to group at 5% level of significance.