Download Chapter 9: Sampling Distributions

Chapter 9: Sampling Distributions
9.1 Sampling Distributions
1. Explain the difference between a parameter and a statistic? A parameter is a number that describes the
population. A parameter is a fixed number, but in practice we do not know its value because we cannot
examine the entire population. A statistic is a number that describes a sample. The value of a statistic is
known when we have taken a sample, but it can change from sample to sample. We often use a statistic to
estimate an unknown parameter.
2. Explain the difference between and . is the mean of a population. is the mean of a sample. would almost certainly take a different value if we chose another sample from the same population.
3. Explain the difference between p and . p represents a population proportion. proportion. The sample proportion estimates the unknown parameter p.
represents the sample
4. What is sampling variability? Sampling variability refers to the fact that the value of a statistic varies in
repeated random sampling.
5. What is meant by the sampling distribution of a statistic? The sampling distribution of a statistic is the
distribution of values taken by the statistic in all possible samples of the same size from the same population.
6. When is a statistic considered unbiased? A statistic used to estimate a parameter is unbiased if the mean of its
sampling distribution is equal to the true value of the parameter being estimated.
7. How is the size of a sample related to the spread of the sampling distribution? Larger samples give smaller
spread, less variability.
8. Discuss the variability of a statistic. The variability of a statistic is described by the spread of its sampling
distribution. This spread is determined by the sampling design and the size of the sample. As long as the
population is much larger than the sample (say, at least 10 times as large), the spread of the sampling
distribution is approximately the same for any population size.
9. At the beginning of this section they said the reasoning of statistical inference rests on asking
a question. What was that question? “How often would this method give a correct answer if I used it very
many times?”
9.2 Sample Proportions
1. In an SRS of size n, what is true about the sampling distribution of when the sample size n increases? As n
increases the sampling distribution of becomes approximately normal.
2. In an SRS of size n, what is the mean of the sampling distribution of ? Is an unbiased estimator for p? The mean of the sampling distribution of is exactly p. Therefore is an unbiased estimator for p.
3. In an SRS of size n, what is the standard deviation of the sampling distribution of ? The standard deviation
of the sampling distribution of is 4. What happens to the standard deviation of as the sample size n increases? The standard deviation of gets
smaller as the sample size n increases because n appears in the denominator of the formula. Therefore is
less variable in larger samples
5. When does the formula apply to the standard deviation of least 10 times as large as the sample. (10n population)
? ONLY when the population is at
6. When the sample size n is large, the sampling distribution of is approximately normal. What test can you
use to determine if the sample is large enough to assume that the sampling distribution is approximately
normal? We will use the normal approximation to the sampling distribution of for values of n and p that
satisfy . This rules out cases where the probability p is very close to 0 or to 1.
7. What two assumptions do we have to make to use the normal approximation to the sampling distribution of ? (The answer to this will be related to your answers to 5 and 6 above.)
1. 10n population 2. 9.3 Sample Means
1. The mean and standard deviation of a population are parameters. What symbols are used to represent these parameters?
Remember: Parameters describe
Populations and
Statistics describe
Samples!!
2. The mean and standard deviation of a sample are statistics. What symbols are used to represent these statistics?
3. Because averages are less variable than individual outcomes, what is true about the standard deviation of the sampling distribution of ? It is smaller than the standard deviation of the population. The
larger the sample, the smaller the standard deviation of the sampling distribution of is, there is less
variability.
4. What is the mean of the sampling distribution of , if is the mean of an SRS of size n drawn from a large population with mean and standard deviation ? Is an unbiased
estimator for he mean of the sampling distribution of is estimator for 
. is an unbiased
5. What is the standard deviation of the sampling distribution of , if is the mean of an SRS of size n drawn from a large population with mean and standard deviation ? he standard deviation of
the sampling distribution of is . 6. To cut the standard deviation of in half, you must take a sample four times as large.
7. When should you use to calculate the standard deviation of ? You should ONLY use this formula
for the standard deviation of if the population is at least 10 times as large as the sample.
8. What assumptions do we have to make to use the normal approximation to the sampling
distribution of ? We can use the normal approximation if the original population is normal, or if our
sample, n, is large. We always need 10n population.
9. State the Central Limit Theorem. Draw an SRS from any population whatsoever with mean and finite
standard deviation . When n is large, the sampling distribution of the sample mean is close to the normal
distribution with mean and standard deviation .
10. What does the Central Limit Theorem say about the shape of the sampling distribution of ?
The CLT says that the distribution of sample means, , from a strongly nonnormal population becomes
more normal as the sample size increases.
11. What does the Central Limit Theorem allow us to do? The CLT allows us to use normal probability
calculations to answer questions about sample means from many observations even when the population
distribution is not normal.