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Material Type: Notes; Subject: Animal Science; University: University of Maryland; Term: Fall 2002;
Typology: Study notes
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1.1 POINT ESTIMATES โ a single number stated as an estimate of some quantitative property of the population
1.2 INTERVAL ESTIMATES โ a statement that a population parameter has a value lying between two specified limits
1.3 CON FIDENCE INTERVA L ESTIMATES โ an interval estimate constructed such that repeated application results in inclusion of the population parameter at a known level of probability
2.1 Given D ISTRIBUTION OF OB SERV ATION S (Yi) which are normally and independently distributed [NID(:, F)].
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Now draw ALL possible samples size n = 4
2.3 CENTRAL LIMIT THEOREM - as sample size increases, the distribution of the means of samples drawn from any distribution will approach the normal distribution with
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3.1 A normal distribution with :=1 and F =1 is known as the stan dard normal distribution, Z represents o bservations from the standard normal. Z for an observation: Y
which states that an observation's deviation from the population mean (:) divided by the population standard deviation (F) is distributed as Z.
3.2 Find the value of Z such that the area from : - Z to : + Z equals 95% of the population, i.e.
| | | | 2.5% | | | 2.5% | | |
which is known as the 95% confidence interval for the population parameter : where:
= confidence coefficient
= LCL = lower confidence limit
= UCL = upper confidence limit
= confidence interval length
4.1 As the confidence coefficient is increased, the interval length increases. Thus the increase in reliability is at expense of usefulness, since interval is less specific about the population value.
Given the same sample of n = 20 from a population where F = 10. Find 99% confidence interval for ::
(n = 20, = 48.94)
CONFIDENCE THAT INTERVAL IS CORRECT ( 1-" )
RISK THAT INTERVAL IS INCORRECT ( " )
LCL UCL INTERVAL LENGTH
95% 5% 44.56 53.32 8. 99% 1% 43.20 54.68 11.
5.1 ROUGH RULE โ a 4 fold increase in sample size results in a decrease in interval length by approximately 1/2.
Now suppose we draw samples of size n = 20, 40 and 80 from a population where F = 10 and that for each sample = 48..
95% CI for each
n LCL UCL INTERVAL LENGTH
20 , 44.56 53.32 8.76 ,
40 * 4x 45.84 52.04 6.20 * .5x
80 - 46.75 51.13 4.38 -
6.3 Rewriting the confidence intervals using t when F is unknown.
where: " = 1 - confidence coefficient
confidence coefficient = 1 - "
Draw a sample o f: n = 20, = 48.94, S = 10.
7.1 GENERAL FOR MULA for normal or approximately normally distributed statistics:
statistic - t",df * (standard error of statistic) < parameter < statistic + t",df * (standard error of statistic)
44.20 # : # 53.68 with 95% confidence
Notice that the interval is longer than the one computed in 6.5.3. This is due to the uncertainty associated with the sample estimate of the standard deviation.
7.2.6 UNPAIRED EQUAL VAR IANC ES: n 1 = n 2 or n 1 ร n2 when F 1 = F 2
where df = (n 1 - 1) + (n 2 - 1)
7.2.7 UNEQUA L VARIAN CES: n 1 = n 2 or n 1 ร n 2 when F 1 ร F 2
7.3 EXAMPLE: find a 95% confidence interval for a coefficient of variation (CV) :
7.4 SIGNIFICANT FIGURES FOR REPORTING STATISTICS: The standard error of a statistic (or confidence limits for a statistic) is the most commonly available measure of the precision of the estimation of the statistic. The nu mber of significant figures reported for a statistic should reflect precision of its estimation. From experience, in most biological research 2 or 3 significant figures is all that can be justified. It is unusual to find more than 3 significant figures, while it is not uncommon to find cases where the precision is so poor that only one significant figure is appropriate.
Several authors have suggested guidelines for reporting significant figures for statistics. I generally use a simple rule that I believe is satisfactory for reporting most statistical results. Remember that these rules are for reporting statistics. Numbers and/or statistics used to compute other statistical values must carry more digits to av oid the accumulation of rounding error.
8.5 formula for P^2 distribution:
8.1 DEFINITION: P^2 is a probability density function which describes the ratio of sample SS to the population variance.
8.2 unlike the normal and t distributions, P^2 cannot take on negative values
8.3 like the t distribution, P^2 is a series of distributions which varies according to degrees of freedom
8.4 P^2 is greatly skewed to right with only 1 df, but as df increases, skewedness decreases
8.6 confidence interval for population parameter: F^2
