econ

Homework 2 Due Tuesday August 17th at 9:00 PM Pacific Time

Note: You must show all work to receive full credit. Where applicable, please round to four decimal places

Name: Student ID: Other Students worked with:

1. Calculate the 95% Z-interval or t-interval for the indicated parameter: (a) µ: X̄ = 240,s2 = 5184,n = 20

(b) µ: X̄ = 325,σ2 = 196,n = 121

(c) p : p̂ = 0.29,n = 35

(d) p : p̂ = 0.82,n = 67

(e) µ: X̄ = 157,s2 = 169,n = 21

(f) p : p̂ = 0.25,n = 25

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2. Find the appropriate percentile (i.e. critical value) from a t-distribution for con- structing the following confidence intervals:

(a) 90% t-interval with n=16

(b) 95% t-interval with n=9

(c) 99% t-interval with n=4

(d) 95% t-interval with n=27

(e) 90% t-interval with n=5

(f) 99% t-interval with n=18

(g) 95% t-interval with n=7

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3a. In a sample of 75 Americans, 54 could not find North Korea on a map. Construct a 97% confidence interval for the true proportion of Americans who cannot find North Korea on a map. Interpret this interval.

(b) Consider a scenario where an individual conducts a follow-up study in one-year. Suppose this individual would like to control the margin of error to be no greater than 5% when he constructs a 97% confidence interval using the follow-up study data. Using p̂ from part (a), what is the minimum sample size required so that the margin of error is no greater than 5%?

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4. There is a group of 80 people all with a fair coin. They all flip their coins 200 times and build a 95% confidence interval for their coin’s true proportion of heads. On average, how many of the 80 people will have confidence intervals that contain the true proportion of 0.5?

5. IQ scores are normally distributed. The average IQ is 100 with a standard devia- tion of 16. If we take a sample of 16 people, what is the probability that the sample average will be larger than 110?

6. Suppose Xi iid∼ Gamma(α,β) for i ∈{1, 2, ...,n}. The mean of a gamma random variable

is E[Xi] = µ = α β

and variance V ar[Xi] = σ 2 = α

β2 . Assume the sample size/normality

condition holds. What is the sampling distribution of X̄ in terms of α and β?

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7. State whether each of the following statements is true or false. If the statement is false, briefly explain why you think it is false.

(a) Ninety-five percent of z-intervals have the form of a statistic plus or minus more than 3 standard errors of the statistic.

(b) All things the same, a 90% confidence interval is shorter than a 95% confidence interval.

(c) By increasing the sample size from n=100 to n=400, we can reduce the margin of error by 50%.

(d) If we double the sample size from n=50 to n=100, the length of the confidence interval is reduced by half.

(e) If the 95% confidence interval for the average Math GRE score of graduating UCI students is 148 to 163, then 152 is a plausible value for the population mean at this level of confidence.

(f) If the 95% confidence interval for the number of moviegoers who purchase popcorn from the concession stand is 25% to 40%, then fewer than half of all moviegoers do not purchase popcorn from the concession stand.

(g) The 95% t-interval for µ only applies if the sample data are nearly normally distributed.

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8. A news reports summarizes a poll of voters and then adds that the margin of error is plus or minus 4%. Explain what this means.

9. What is more likely to contain µ, the z-interval X̄ ± 1.96σ/ √ n or the t-interval

X̄ ± t0.025,n−1S/ √ n?

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