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Belief in the Dream: Analyzing Z-scores & Sample Sizes
Chapter , Problem 6.2(choose chapter or problem)
About 77 % of young adults think they can achieve the American dream. Determine if the following statements are true or false, and explain your reasoning.
(a) The distribution of sample proportions of young Americans who think they can achieve the American dream in samples of size 20 is left skewed.
(b) The distribution of sample proportions of young Americans who think they can achieve the American dream in random samples of size 40 is approximately normal since \(n \geq 30\).
(c) A random sample of 60 young Americans where 85% think they can achieve the American dream would be considered unusual.
(d) A random sample of 120 young Americans where 85% think they can achieve the American dream would be considered unusual.
Questions & Answers
QUESTION:
About 77 % of young adults think they can achieve the American dream. Determine if the following statements are true or false, and explain your reasoning.
(a) The distribution of sample proportions of young Americans who think they can achieve the American dream in samples of size 20 is left skewed.
(b) The distribution of sample proportions of young Americans who think they can achieve the American dream in random samples of size 40 is approximately normal since \(n \geq 30\).
(c) A random sample of 60 young Americans where 85% think they can achieve the American dream would be considered unusual.
(d) A random sample of 120 young Americans where 85% think they can achieve the American dream would be considered unusual.
ANSWER:Step 1 of 4
(a)
The statement is True.
In the success-failure condition we expected to see at least 10 successes and 10 failures in our sample, i.e. 
Population proportion, p = 77% = 0.77 (Given)
sample of size, n = 20 (Given)
Then, np = 20(1 - 0.77) = 4.8
So, np < 10, this statement doesn’t satisfy the success-failure condition. So it’s not going to be a Normal distribution. Since the mean is 0.77, which is above 50% of the population mean, and the sample size is small so the standard deviation will be high. This will result in a left skewed distribution,