16. people are poor at making judgments about probabil-ity. one source of error in judgments of probability…

16. people are poor at making judgments about probabil-ity. one source of error in judgments of probability isthe base rate fallacy in which people ignore the baserates of low probability events. in a study of the baserate fallacy by bar - hillel (1980), participants wereexposed to a vignette about a traffic accident. in thescenario, a taxicab was observed in a hit - and - run ac -cident. in the city where the accident occurred, 85%of cabs are blue and 15% of cabs are green. later, awitness testified that the cab in the accident was greenand the witness was shown to be 80% accurate inidentifying blue and green cabs (i.e., 20% of the time,the witness confused the cabs). what do you think isthe probability that a green cab was in the hit - and - run?most participants who encounter this problem reportthat the probability of the cab being green is muchhigher than the actual probability of 41%. that is,most participants ignore the fact that green cabs arerelatively rare. suppose that a researcher replicatesthe bar - hillel experiment with a sample of ( n = 16 )participants. the researcher observes an average ratedprobability of ( m = 60.06% ) with ( ss = 656.66 ).a. use a two - tailed test ( (alpha =.05) ) of the hypothesisthat participants showed a base rate fallacy. assumethat ( mu = 41 ) if there is no base rate fallacy.b. compute two different measurements of effect size.

16. people are poor at making judgments about probabil-ity. one source of error in judgments of probability isthe base rate fallacy in which people ignore the baserates of low probability events. in a study of the baserate fallacy by bar - hillel (1980), participants wereexposed to a vignette about a traffic accident. in thescenario, a taxicab was observed in a hit - and - run ac -cident. in the city where the accident occurred, 85%of cabs are blue and 15% of cabs are green. later, awitness testified that the cab in the accident was greenand the witness was shown to be 80% accurate inidentifying blue and green cabs (i.e., 20% of the time,the witness confused the cabs). what do you think isthe probability that a green cab was in the hit - and - run?most participants who encounter this problem reportthat the probability of the cab being green is muchhigher than the actual probability of 41%. that is,most participants ignore the fact that green cabs arerelatively rare. suppose that a researcher replicatesthe bar - hillel experiment with a sample of ( n = 16 )participants. the researcher observes an average ratedprobability of ( m = 60.06% ) with ( ss = 656.66 ).a. use a two - tailed test ( (alpha =.05) ) of the hypothesisthat participants showed a base rate fallacy. assumethat ( mu = 41 ) if there is no base rate fallacy.b. compute two different measurements of effect size.

Answer

Explanation:

Step1: Calculate sample variance ($s^{2}$)

Sample variance formula: $s^{2}=\frac{SS}{n - 1}$. Given $SS = 656.66$ and $n=16$, then $s^{2}=\frac{656.66}{16 - 1}=\frac{656.66}{15}=43.7773$.

Step2: Calculate standard error ($s_{M}$)

Standard error formula: $s_{M}=\frac{s}{\sqrt{n}}$, and since $s=\sqrt{s^{2}}$, $s=\sqrt{43.7773}\approx6.616$. Then $s_{M}=\frac{6.616}{\sqrt{16}}=\frac{6.616}{4}=1.654$.

Step3: Calculate t - statistic

t - statistic formula: $t=\frac{M-\mu}{s_{M}}$. Given $M = 60.06$, $\mu = 41$, then $t=\frac{60.06 - 41}{1.654}=\frac{19.06}{1.654}\approx11.52$.

Step4: Determine critical t - values

For a two - tailed test with $\alpha = 0.05$ and $df=n - 1=16 - 1 = 15$, from the t - distribution table, the critical t - values are $t_{\alpha/2,df}=\pm 2.131$.

Step5: Make a decision

Since $|t|=11.52>2.131$, we reject the null hypothesis.

Answer:

We reject the null hypothesis that there is no base rate fallacy.