Greece has faced a severe economic crisis since the end of 2009. A Gallup poll s
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Question
Greece has faced a severe economic crisis since the end of 2009. A Gallup poll surveyed 1,000 randomly sampled Greeks in 2011 and found that 25% of them said they would rate their lives poorly enough to be considered "suffering." a) Define the population parameter of interest. What is the point estimate for this parameter? b) Check that the conditions required for consulting a Confidence Interval based on these data are met. c) Construct a 95% Confidence Interval for the proportion of Greeks who are "suffering." d) Describe what would happen to this Confidence Interval if we decided to use a higher confidence level. e) Describe what would happen to the Confidence Level if we used a larger sample.Explanation / Answer
a) The parameter in interest is the proportion of all Greeks who are suffering from the economic crisis.
The point estimate, p = 25% = 0.25
b) The sampling method used is simple random sampling.
Each sample point can result in just two possible outcomes. Here it is either suffering or not suffering
At least 10 successes and 10 failures. 25% of 1000 people = 250 people said suffering and 750 said not suffering.
The population size is at least 20 times as big as the sample size. The population of Greeks run into millions and it is safe to assume this condition is satisfied.
c) Standard error, SEp = sqrt[ p * ( 1 - p ) / n ],
where p is the sample proportion, and n is the sample size.
SEp = sqrt[0.25*(1-0.25)/1000] = 0.0136931
zcrit = 1.96 for = 0.05 [since the sample size is so large, z score can be used]
lower bound = p - zcrit*SE = 0.25 - 1.96*0.0136931 = 0.223
upper bound = p + zcrit*SE = 0.25 + 1.96*0.0136931 = 0.277
d) For a higher confidence interval, the interval will become wider to accomodate more values. Therefore our confidence is higher that the population proportion falls within a wider interval.
e) Larger sample sizes decrease the value of the standard error and in turn the margin of error decreases too. This causes the confidence interval to be narrower for the same confidence coefficient.
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