QSCI381: Introduction to Probability and Statistics

Problem set #5

1 a. What is the probability of selection of each possible sample of size 2 from a population of size 7?

b. Select all possible samples of size 2 from the population of 7 digits (1,...,7). Compute mean and variance for each of the samples and show that the average of all sample means equals the population mean.

c. Verify that the population variance {s 2 = }, (computed with N-1 as the divisor) equals the average of all sample variances.

d. Compute the variance of sample means in b) above. Show that it is close to the variance calculated from the population variance, sample size and the Finite Population Correction factor.

2. Judy’s doctor is concerned that she may suffer from hypokalemia (low potassium in the blood). There is variation both in the actual potassium level and in the blood test that measures the level. Judy’s potassium level varies according to the normal distribution with m = 3.8 and s = 0.2. A patient is classified as hypokalemic if the potassium level is below 3.5.

  1. If a single potassium measurement is made, what is the probability that Judy is diagnosed hypokalemic?
  2. If measurements are made instead on 4 separate days and the mean result is compared with the criterion 3.5, what is the probability that Judy is diagnosed as hypokalemic?

3 The mean of a random sample of size 100 is used to estimate the mean of a very large population, which has a standard deviation of 2.5 kg. If we use the central limit theorem, what can we assert about the probability that the error (i.e., the difference between the population mean and the observed sample mean) will be?

(a) less than 0.5 kg

(b) not more than 0.36 kg

(c) at least 0.45 kg
  1. The number of accidents per week at a hazardous intersection varies with mean 2.2 and S.D. 1.4.
  1. Let be the mean number of accidents at the intersection during a year (52 weeks). What is the approximate distribution of according to the central limit theorem?
  2. What is the approximate probability that is less than 2?
  3. What is the approximate probability that fewer than 100 accidents per year (52 weeks)?

5 Select 100 simple random samples, each of size 25 (using EXCEL or MINITAB), from the following probability distributions:

Poisson µ = 1.5;

Binomial n = 40 & p = 0.25. (m = 10, s 2 = 7.5)

 

Do the following for each of these distributions:

 

a. Compute means of 100 samples of size 25. Find the overall mean of these sample means. How far is this overall mean from the theoretical mean?

b. Compute the standard deviation of the 100 sample means and compare it with the theoretical standard error for the given sample size.

c. Compute the percentage of sample means within 2 theoretical standard errors of the true mean. Compare this with 95% for normal distribution

d. Draw histogram of the 100 sample means from the three distributions. How does the shape of the distribution compare with the shape of the normal distribution?