Statistics

Instructions: Each problem/question is of equal value (2 points). You solve problems with closed

books, notes, etc. Do not forget accurately write down proofs for problems 9 and 10.

1. A biologist studies intensities of 16 probs in a particular microarray. It is assumed that the

intensities are samples from a normal distribution with a given population variance 49. The observations

yielded the sample mean intensity X̄ = 28 and the sample standard deviation 5. Find a 99% confidence

interval for the population mean.

2. Twelve randomly selected citrus trees have a sample mean height of 13.8 feet with a known

population standard deviation 1.2 feet. Twenty randomly selected apple trees have a sample mean

height of 12 feet with a known population standard deviation of 1.5 feet. Assuming that the two

populations are independent and normal, construct a 95% confidence interval for the difference between

the population means.

3. Assume that the age of trees on two different farms has the same variance. For the first farm, a

sample of 8 trees gave the sample mean age 7 and the sample variance 3, while for the second farm a

sample of 9 trees gave the sample mean 8 and the sample variance 4. Find a 98% confidence interval

for the difference between the population mean ages.

4. A sample survey at a supermarket showed that 200 of 300 shoppers regularly use coupons.

Construct a 95% confidence interval for the corresponding true proportion.

5. In a restaurant A only 68 of 200 patrons ordered dessert, while in restaurant B 97 of 240

patrons ordered dessert. Construct a 94% confidence interval for the difference between the underlying

population proportions.

6. A sample of 21 cars has shown that the sample variance of their speeds is 15. Assuming that the

speeds are normally distributed, construct a 98% confidence interval for the population variance.

7. A sample A of size 21 produced a sample variance 9. A sample B of size 36 produced a sample

variance 16. Assume that the samples are independent and the underlying populations are normal.

Find a 98% confidence interval for the ratio of the population variances.

8. Let a sample of size 12 is taken from a normal population. The sample statistics are: sample

mean is 5 and sample variance is 14. Find a 95% confidence interval for the population mean.

9. Consider an F(k,m) distribution. Prove that fβ,k,m = 1/f1−β,m,k.

10. Consider the case of two samples of sizes n1 < 10 and n2 < 20 from two independent normal

populations with the same variance. Suppose that the corresponding sample means are X̄1 and X̄2,

and the corresponding sample variances are S21 and S 2 2 . Write down a 1 − α confidence interval for the

difference between population means and then prove that this is indeed the desired confidence interval.

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