In a 1997 study, people were observed for about 10 seconds in public places, suc
ID: 3220667 • Letter: I
Question
In a 1997 study, people were observed for about 10 seconds in public places, such as malls and restaurants, to determine whether they smiled during the randomly chosen 10 second interval. The table shows the results for comparing males and females. (Source: M. S. Chappel, Frequency of public smiling over the life span, Perceptual and Motor Skills, vol. 45: 474, 1997) a. Find and compare the sample percentage of women who were smiling and men who were smiling. b. Treat this as though it were a random sample, and test whether there are differences in the proportion of men and the proportion of women who smile. c. Explain why there is such a small p-value even though there is such a small difference in sample percentages.Explanation / Answer
a) total number of females, nw = (4471+4278) = 8749
number of females who smile, xw = 4471
Proportion of females who smile, pw = 4471/8749 = 0.511
total number of males, nm = (3269+3806) = 7075
number of males who smile, xm= 3269
Proportion of males who smile, pm = 3269/7075 = 0.462
There is a higher proportion of females who smile in the random 10 second span than males.
b) Hypothesis testing, H0: pw - pm = 0, Ha: pw - pm 0
z = (pw - pm)/sqrt(p*(1-p*)((1/nw) + (1/nm))), where p* = (xw + xm)/(nw+nm) = (4471+3269)/(8749+7075) = 0.489130
z = (0.511-0.462)/sqrt(0.489130*0.510870*((1/8749)+(1/7075))) = 6.131
p = 0, The result is statistically significant and there is sufficient evidence to suggest that there is difference between the proportions of males and females who smile in the 10-second span.
c) The very small p value is due to the large sample size that was used. Due to the large sample size, the standard error or the denominator of the z value is very small , resulting in a large z value.
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