ESSENTIALS OF STATISTICS-W/APLIA
ESSENTIALS OF STATISTICS-W/APLIA
4th Edition
ISBN: 9781305611177
Author: HEALEY
Publisher: CENGAGE L
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Chapter 10, Problem 10.14P

S O C A researcher is concerned with the relationship between attitudes toward violence and violent behavior. Do attitudes "cause” behavior? Do people who have positive attitudes toward violence have higher rates of violent behavior? A pretest was conducted on 70 respondents; among other things, the respondents were asked, "Have you been involved in a violent incident of any kind over the past six months?" The researcher established the following relationship:

Attitude Toward Violence
Involvement Favourable Unfavourable Totals
Yes 16 19 35
No
Totals
14 30 21 40 35 70

The chi square for this table is 0.23, which is not significant at the 0.05 level (confirm this with your own calculations). Undeterred by this result, the researcher proceeded with the project and gathered a random sample of 7000. In terms of percentage distributions, the results for the full sample were exactly the same as for the pretest:

Attitude Toward Violence
Involvement Favourable Unfavourable Totals
Yes 1600 1900 3500
No
Totals
1400 3000 2100 4000 3500 7000

However, the chi square obtained is a very healthy 23.4 (confirm with your own calculations). Why is the full-sample chi square significant when the pretest was not? What happened? Do you think that the second result is important?

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iid 1. The CLT provides an approximate sampling distribution for the arithmetic average Ỹ of a random sample Y₁, . . ., Yn f(y). The parameters of the approximate sampling distribution depend on the mean and variance of the underlying random variables (i.e., the population mean and variance). The approximation can be written to emphasize this, using the expec- tation and variance of one of the random variables in the sample instead of the parameters μ, 02: YNEY, · (1 (EY,, varyi n For the following population distributions f, write the approximate distribution of the sample mean. (a) Exponential with rate ẞ: f(y) = ß exp{−ßy} 1 (b) Chi-square with degrees of freedom: f(y) = ( 4 ) 2 y = exp { — ½/ } г( (c) Poisson with rate λ: P(Y = y) = exp(-\} > y! y²
2. Let Y₁,……., Y be a random sample with common mean μ and common variance σ². Use the CLT to write an expression approximating the CDF P(Ỹ ≤ x) in terms of µ, σ² and n, and the standard normal CDF Fz(·).
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