Sampling Distributions Solved Problems for Beginning Statistics
Review the following concepts if necessary:
- Simple Random Sampling for Beginning Statistics
- Sampling Distributions for Beginning Statistics
- Mean and Standard Deviation of the Sample Mean for Beginning Statistics
- Shape of the Sampling Distribution of the Sample Mean and the Central Limit Theorem for Beginning Statistics
- Sampling Distribution of the Sample Proportion for Beginning Statistics
- Shape of the Sampling Distribution of the Sample Proportion and the Central Limit Theorem for Beginning Statistics
Simple Random Sampling
- The top nine medical doctor specialties in terms of median income are as follows: radiology, surgery, anesthesiology, obstetrics/gynecology, pathology, internal medicine, psychiatry, family practice, and pediatrics. How many simple random samples of size 4, chosen without replacement, are possible when selected from the population consisting of these nine specialties? What is the probability associated with each possible simple random sample of size 4 from the population consisting of these nine specialties?
- In Problem 7.1, how many of the 126 possible samples of size 4 would include the specialty pediatrics?
Ans: The number of possible samples is equal to = 126. Each possible sample has probability = .00794 of being selected.
Ans: The number of samples of size 4 which include the specialty pediatrics would equal the number of ways the other three specialties in the sample could be selected from the other eight specialties in the population. The number of ways to select 3 from 8 is = 56. One such sample is the sample consisting of the following: (radiology, surgery, pathology, pediatrics). There are 55 other such samples containing the specialty pediatrics.
Using Random Number Tables
- Use Table 7.1 of this chapter to obtain a sample of size 5 from the population consisting of the 50 states and the District of Columbia. Assume that the 51 members of the population are listed in alphabetical order as shown in Table 7.11. Start with the two digits in columns 11 and 12, and line 1 of Table 7.1. Read down these two columns until five unique states have been selected. Give the numbers and the selected states.
- The random numbers in Table 7.1 are used to select 15 faculty members from an alphabetical listing of the 425 faculty members at a university. The total faculty listing goes from 001: Lawrence Allison to 425: Carol Ziebarth. A random start position is determined to be columns 31 through 33 and line 1. Read down these columns until you reach the end. Then go to columns 36 through 38, line 1 and read down these columns until you reach the end. If necessary, go to columns 41 through 43 and line 1 and read down these columns until you obtain 15 unique numbers. Give the 15 numbers which determine the 15 selected faculty members.
Ans: The random numbers are 44, 12, 24, 10, and 07. From an alphabetical listing of the 50 states and the District of Columbia in Table 7.11, the following sample is obtained: (Connecticut, Florida, Hawaii, Minnesota, and Texas)
Ans: The numbers, in the order obtained from Table 7.1 are: 345, 254, 160, 105, 283, 026, 099, 385, 157, 393, 013, 126, 110, 004, and 279.
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