Review 3 - NAU jan.ucc.nau.edu web server
Review 3
Chapter 8
1. A statistic ( any quantity computed from values in a sample (for example, [pic], s, the sample median, the sample interquartile range and so on).The distribution of a statistic is called its sampling distribution.
A population parameter ( any quantity computed from values in a population (for example, (, (, the population median, the population interquartile range and so on).
• The difference between a statistic and a population parameter.
(1) A statistic is a sample characteristic, whereas a population parameter is a population characteristic.
(2) The observed value of a statistic varies from sample to sample. However, a population parameter is a fixed number, which is generally unknown.
2. Properties of the sampling distribution of [pic]
Let [pic]denote the mean of the observations in a random sample of size n from a population having mean ( and standard deviation (. Denote the mean value of the [pic] distribution by [pic] and the standard deviation of [pic]distribution by [pic]. Then the following rules hold.
Rule 1: [pic]=(
Rule 2: [pic]=[pic].
Rule 3: When the population distribution is normal, the sampling distribution of [pic]is also normal for any sample size n. Thus, the standardized variable
[pic]
has the standard normal (z) distribution.
Rule 4: (Central Limit Theorem) When n is sufficiently large (n≥30), the sampling distribution of [pic] is well approximated by a normal curve, even when the population distribution is not itself normal. So, the standardized variable
[pic]
has approximately the standard normal (z) distribution.
3. Properties of the sampling distribution of p
Let p be the proportion of S’s in a random sample of size n from a population whose proportion of S’s is (. Denote the mean value of p by (p and the standard deviation of p by (p. Then the following rules hold.
Rule 1: (p = (
Rule 2: [pic]
Rule 3: (Central Limit Theorem) When n is large and ( is not too near 0 or 1 (n( ( 10 and n(1-( ) ( 10), the sampling distribution of p is approximately normal. Thus, the standardized variable
[pic]
has approximately the standard normal (z) distribution.
Chapter 9
4. A point estimate of a population characteristic is a single number computed from sample data and represents a plausible value of the characteristic. A point estimate is obtained by (i) selecting an appropriate statistic; (ii) computing the value of the statistic for the given sample.
A statistic whose mean is equal to the value of the population characteristic being estimated is said to be an unbiased statistic. A statistic that is not unbiased is said biased.
5. Criteria for choosing among competing statistics
a) First we choose an unbiased statistic if there is one.
b) If several unbiased statistics could be used for estimating a population characteristic, we choose the one with the smallest standard deviation.
6. Statistics used to estimate some important population characteristics
|Population characteristic to be estimated |Statistic to use |Unbiasedness |
|Population proportion, ( |p |Unbiased |
|Population mean, ( |[pic] |Unbiased |
|Population variance, (2 |s2 |Unbiased |
|Population standard deviation, ( |s |Biased |
|Population median |Sample median |Unbiased if symmetric |
| | |Biased if skewed |
7. A confidence interval for a population characteristic is an interval of plausible values for the characteristic. It is constructed so that, with a chosen degree of confidence, the value of the characteristic will be captured inside the interval.
The confidence level associated with a confidence interval estimate is the success rate of the method used to construct the interval.
The standard error of a statistic is the estimated standard deviation of the statistic.
If the sampling distribution of a statistic is normal (approximately), the bound on error of estimation, B, associated with a confidence interval is
(z critical value)((standard deviation of the statistic).
8. The large-sample confidence interval for (
When
(1) p is the sample proportion from a random sample, and
2) the sample size n is large (np ( 10 and n(1-p) ( 10)
the general formula for a confidence interval for a population proportion ( is
p ( (z critical value) [pic]
The desired confidence level determines the z critical value. The three most commonly used confidence levels, 90%, 95%, and 99%, use z critical values 1.645, 1.96, and 2.58, respectively.
9. The sample size required to estimate a population proportion ( to within an amount B with a confidence level is
n = ((1-() [pic]
The value of ( may be estimated using prior information. In the absence of any such information, using ( = .5 in this formula gives a conservatively large value for the required sample size.
10. The one-sample z confidence interval for (
When
1. [pic] is the sample mean of a random sample,
2. the population distribution is normal or the sample size n is large (generally n ( 30), and
3. the population standard deviation ( is known
the formula for a confidence interval for a population mean ( is
[pic] ( (z critical value) ([pic])
11. Let x1, x2, (, xn be a random sample from a normal population distribution. Then the probability distribution of the standardized variable
[pic]
has the t distribution with n-1 df.
12. The one-sample t confidence interval for (
When
1. [pic]is the sample mean of a random sample
2. the population distribution is normal or the sample size n is large (generally n ( 30), and
3. the population standard deviation ( is unknown
the formula for a confidence interval for population mean ( is
[pic] ( (t critical value) ([pic])
where the t critical value is based on n-1 df, which can be found by Appendix Table 3 on page 708.
13. The sample size required to estimate a population mean ( to within an amount B with a confidence level is
n =[pic].
If ( is unknown, it may be estimated based on previous information or, for a population that is not too skewed, by using (range)/4.
14. Important examples in the Notes:
Examples: 8.1, 8.2, 8.3, 8.4, 9.1, 9.2, 9.3, 9.4.
15. Exercise in class: A random sample of n = 12 four-year-old red pine trees was selected, and the diameter (in inches) of each tree's main stem was measured. The resulting observations are as follows:
11.3 10.7 12.4 15.2 10.1 12.1 16.2 10.5 11.4 11.0 10.7 12.0
a) Give a point estimate of (, the population mean diameter.
b) Give a point estimate of the population median diameter.
c) Give a point estimate of (, the population proportion of trees whose main stem diameters are at least 12 inches.
d) Compute a point estimate of σ, the population standard deviation of main stem diameter.
e) Suppose that the diameter distribution is normal. Then the 90th percentile of the diameter distribution is μ+1.28σ (so 90% of all trees have diameters less than this value). Compute a point estimate for this percentile.
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