The following data shows the ages of a sample of 10 US billionaires; use it to c

Question

The following data shows the ages of a sample of 10 US billionaires; use it to complete the following:

a) Find the best point estimate of the average age of all US billionaires. Show your work/calculations

b) Construct a 95% confidence interval of the average age of all US billionaires. Show your work/calculations

c) Why can't we answer the following from the given data? How large a sample would necessary if we want to estimate the true average age of all US billionaires to within 9 years with 99% confidence?

d) Suppose the average net worth of all US billionaires is $8.65 billion and standard deviation= $4.3 billion. Assume that net worth is normally distributed.

1) If one US billionaire is selected at random, determine the probability that their net worth is more thatn $10.8 billion. Show all relevant work.

2) If 25 US billionaires are selected at random, determine the probability that their net worth is more than $10.8 billion

Age Net Worth (billions) 56 18 39 14 42 12 60 14 84 11 37 10 68 10 66 7 73 7 55 5

Explanation / Answer

a) Find the best point estimate of the average age of all US billionaires. Show your work/calculations

Answer : Best point estimate of the average age of US billionaaires = x? = 58 years

b) Construct a 95% confidence interval of the average age of all US billionaires. Show your work/calculations

Answer : Here sample standard deviation s = 15.42

standard error of sample mean = s/sqrt(n) = 15.42/ sqrt(10) = 4.876

95% confidence interval = x? +- tdF,0.05 = 58 +- 2.2652 * 4.876 = (46.97 year, 69.03 year)

c) Why can't we answer the following from the given data? How large a sample would necessary if we want to estimate the true average age of all US billionaires to within 9 years with 99% confidence?

Answer : As here population standard deviation is not given so we cannot calculate the margin of error here otherwise we have to take point estimate of sample standard deviation as standard deviation.

d) Suppose the average net worth of all US billionaires is $8.65 billion and standard deviation= $4.3 billion. Assume that net worth is normally distributed.

1) If one US billionaire is selected at random, determine the probability that their net worth is more thatn $10.8 billion. Show all relevant work.

Here if x is the Net worth of any random US billionaire

Pr(x > $ 10.8 billion) = Pr(x > $ 10.8 billion ; 8.65 billion ; 4.3 billion)

Z = (10.8 - 8.65)/4.3 = 0.5

Pr(x > $ 10.8 billion) = Pr(x > $ 10.8 billion ; 8.65 billion ; 4.3 billion) = 1- Pr(Z < 0.5) = 1 - 0.6915 = 0.3085

2) If 25 US billionaires are selected at random, determine the probability that their net worth is more than $10.8 billion

Here standard error of sampling distribution = 4.3/sqrt(25) = $ 0.86 billion

Pr(x? > $ 10.8 billion) = Pr(x? > $ 10.8 billion ; 8.65 billion ; 0.86 billion)

Z = (10.8 - 8.65)/0.86 = 2.5

Pr(x? > $ 10.8 billion) = Pr(x? > $ 10.8 billion ; 8.65 billion ; 0.86 billion) = 1- Pr(Z < 2.5) = 1 - 0.9938 = 0.0062

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