Sampling Distributions Questions and Answers

Answer: B

The correct statements are (i), (ii) and (iv).

The size of a sample depends mainly on the precision required and on how variable the population is. (i) is true: the precision needed in the estimates is a key factor. (ii) is true: the standard error of a mean falls as \(\sigma/\sqrt{n}\), so greater precision needs a larger sample. (iv) is true: the more heterogeneous the population, the larger its variance, and the larger the sample needed for the same precision.

(iii) is false, since it says the opposite of (ii): a smaller sample gives a larger standard error and therefore less precision. Every option containing (iii) is therefore wrong.

Answer: A

The sample size is 25, option A. The standard error of the mean is the population standard deviation divided by the square root of the sample size.

\[ \begin{aligned} SE &= \frac{\sigma}{\sqrt{n}} \\ 4 &= \frac{20}{\sqrt{n}} \\ \sqrt{n} &= 5 \\ n &= 25 \end{aligned} \]

Option C, 5, is the value of the square root and is what you get if you stop one step early. Option B, 80, comes from multiplying 20 by 4, and option D, 100, from taking n = 5 squared times 4. A quick check: 20 divided by the square root of 25 is 20 divided by 5, which is 4.

Answer: D

Judgement sampling is not a random sampling method, option D. In judgement (purposive) sampling the researcher chooses the units by personal judgement, so units do not have a known chance of selection and the sampling error cannot be computed in the usual way.

Stratified sampling (option A) draws random samples from each stratum. Cluster sampling (option B) randomly selects whole groups. Systematic sampling (option C) takes every k-th unit after a random start, and it is a probability method because the start is random. All three use chance in selection, and judgement sampling does not.

Answer: D

Stratified sampling is a random (probability) method: the population is divided into strata and a random sample is drawn from each, so every unit has a known chance of selection. Judgement sampling relies on the researcher's choice of typical units, quota sampling fills fixed numbers from groups without random selection, and convenience sampling takes whatever units are easy to reach. These three are non-probability methods, so the answer is stratified sampling.

Answer: C

A large sample size is expected to give a high level of precision. As the sample grows, the standard error falls roughly in proportion to one over the square root of the sample size, so estimates are closer to the true value.

A low level of precision is the effect of a small sample. A larger sample does not increase the sampling error, and it does not change the standard deviation of the population. The answer is option C.

Answer: C

A population from which samples are drawn is called the accessible population, option (C), as the key takes it. The group of subjects that the researcher can actually reach, and from which the sample is chosen, is the accessible population. The target population (B) is the wider group to which the results are meant to apply, and the total population (A) and the universal population (D) are the whole group and not the one that is sampled from.

Answer: A

The t-distribution is approximately normal for a sample size of 30 or more, option (A). As the sample size, and so the degrees of freedom, grow, the t-distribution comes closer to the normal distribution, and by about thirty observations the difference is small.

For samples of 10, 15 or 20 the tails of the t-distribution are still noticeably heavier than those of the normal curve, so the t-values differ appreciably from the normal ones.

A rule of thumb in statistics is that a sample of 30 or more is large enough to use the normal distribution in place of the t-distribution.