Question

The state of New South Wales has an unemployment rate of 5%. The state conducts monthly surveys in order to track the unemployment rate. In a recent month, a random sample of 500 people showed that 20 were unemployed.

1. If the true unemployment rate is 5%, describe the sampling distribution of the sample proportion of unemployed people? (1 pt)
2. Find the probability that the sample unemployment rate is at most 4%. (2 pts)
3. Assume the population proportion p is unknown. Based on the most recent sample, find the probability that the sample proportion will lie within 0.005 of the true proportion pof people who are unemployed. (4 pts)

Answer 1

Answer:

Given Data

Theory : If   is observed proportion of a event out of n samples

Then , by large sample theory (p being the true value)

So inour case , true proportion , p = 0.05

and sample size , n = 500

So, from the above theorem , sampling distribution of   can e described as :-

The above is sampling distribution of sample proportion  

b) Probability that the sample unemployment rate is atmost 4%.

= 0.1524

c) Now , recall that in past (a) , we were given the true value of p

Now here , p is unknown.

So to apply the CLT theory we need estimate of p.

So that we can get sampling distribution of .

From the sample (recent sample) :-

  

= 0.04

Just plug in = 0.04 in sampling distribution.

d)

  

  

  

  

= 0.43169

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