For example: Suppose we compute an interval estimate of a population parameter. Confidence Level The probability part of a confidence interval is called a confidence level. The confidence level describes the likelihood that a particular sampling method will produce a confidence interval that includes the true population parameter.
For Example: Suppose we collected all possible samples from a given population, and computed confidence intervals for each sample. Some confidence intervals would include the true population parameter; others would not. Margin of Error In a confidence interval, the range of values above and below the sample statistic is called the margin of error. Example Which of the following statements is true. When the margin of error is small, the confidence level is high.
When the margin of error is small, the confidence level is low. A confidence interval is a type of point estimate. A population mean is an example of a point estimate. E None of the above. Solution The correct answer is E. The confidence level is not affected by the margin of error. When the margin of error is small, the confidence level can low or high or anything in between. A confidence interval is a type of interval estimate, not a type of point estimate. A population mean is not an example of a point estimate; a sample mean is an example of a point estimate.
Standard Error The standard error is an estimate of the standard deviation of a statistic. This lesson shows how to compute the standard error, based on sample data.
The standard error is important because it is used to compute other measures, like confidence intervals and margins of error. Notation Population parameter N: Number of observations in the population Ni: Number of observations in population i P: Proportion of successes in population Pi: Proportion of successes in population i : Population mean i: Mean of population i : Population standard deviation p: Standard deviation of p Sample statistic n: Number of observations in the sample ni: Number of observations in sample i p: Proportion of successes in sample pi: Proportion of successes in sample i x: Sample estimate of population mean xi: Sample estimate of i s: Sample estimate of SEp: Standard error of p.
Standard Deviation of Sample Estimates Statisticians use sample statistics to estimate population parameters. Naturally, the value of a statistic may vary from one sample to the next. The variability of a statistic is measured by its standard deviation.
Standard Error of Sample Estimates Sadly, the values of population parameters are often unknown, making it impossible to compute the standard deviation of a statistic.
When this occurs, use the standard error. The standard error is computed solely from sample attributes. The standard deviation is computed solely from sample attributes. The standard error is a measure of central tendency. Solution The correct answer is A.
The standard error can be computed from a knowledge of sample attributes - sample size and sample statistics. The standard deviation cannot be computed solely from sample attributes; it requires a knowledge of one or more population parameters. The standard error is a measure of variability, not a measure of central tendency. For Example: Suppose we wanted to know the percentage of adults that exercise daily. We could devise a sample design to ensure that our sample estimate will not differ from the true population value by more than, say, 5 percent the margin of error 90 percent of the time theconfidence level.
How to Compute the Margin of Error The margin of error can be defined by either of the following equations: 1. Otherwise, use the second equation. How to Find the Critical Value 1 The critical value is a factor used to compute the margin of error.
The central limit theorem states that the sampling distribution of a statistic will be normal or nearly normal, if any of the following conditions apply: 1. The population distribution is normal. The sampling distribution is symmetric, unimodal, without outliers. The sampling distribution is moderately skewed, unimodal, without outliers. The sample size is 30 or greater than 30, without outliers.
How to Find the Critical Value 2 When one of these conditions is satisfied, the critical value can be expressed as at score or as a z score. To find the critical value, follow these steps: 1. To express the critical value as a z score, find the z score having a cumulative probability equal to the critical probability.
To express the critical value as a t score, follow these steps: a Find the degrees of freedom DF. When estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one.
For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
Example Nine hundred high school freshmen were randomly selected for a national survey. Among survey participants, the mean grade-point average GPA was 2. Solution The correct answer is B. To compute the margin of error, we need to find the critical value and the standard error of the mean. To find the critical value, we take the following steps: 1.
Find the critical z score. Since the sample size is large, the sampling distribution will be roughly normal in shape. Therefore, we can express the critical value as a z score. For this problem, it will be the z score having a cumulative probability equal to 0. Using the Normal Distribution Tabel, we find that the critical value is 1. Confidence Interval Statisticians use a confidence interval to describe the amount of uncertainty associated with a sample estimate of a population parameter.
How would you interpret this statement? This is incorrect. Like any population parameter, the population mean is a constant, not a random variable. It does not change. The probability that a constant falls within any given range is always 0. How to Interpret Confidence Intervals 2 The confidence level describes the uncertainty associated with a sampling method.
Suppose we used the same sampling method to select different samples and to compute a different interval estimate for each sample. Some interval estimates would include the true population parameter and some would not. Confidence Interval Data Requirements To express a confidence interval, you need three pieces of information. Confidence level 2. Statistic 3. And the uncertainty associated with the confidence interval is specified by the confidence level.
Note: Often, the margin of error is not given; we must calculate it. How to Construct a Confidence Interval 1 There are four steps to constructing a confidence interval: 1. Identify a sample statistic.
Choose the statistic sample mean, sample proportion that you will use to estimate a population parameter. Select a confidence level. As we noted in the previous section, the confidence level describes the uncertainty of a sampling method. How to Construct a Confidence Interval 2 3. Find the margin of error. If you are working on a homework problem or a test question, the margin of error may be given. Often, however, you will need to compute the margin of error, based on one of the following equations.
Specify the confidence interval. The uncertainty is denoted by the confidence level. And the range of the confidence interval is defined by the following equation. Example Suppose we want to estimate the average weight of an adult male in Dekalb County, Georgia. We draw a random sample of 1, men from a population of 1,, men and weigh them.
We find that the average man in our sample weighs pounds, and the standard deviation of the sample is 30 pounds. To specify the confidence interval, we work through the four steps below. Since we are trying to estimate the mean weight in the population, we choose the mean weight in our sample as the sample statistic. In this case, the confidence level is defined for us in the problem. Download Free PDF.
Empirical statistical estimates for sequence similarity searches 1 1 Edited by F. Cohen J Mol Biol, William Pearson. A short summary of this paper. Download Download PDF. Translate PDF. These estimates are derived using the extreme value USA distribution from the mean and variance of the local similarity scores of unrelated sequences after the scores have been corrected for the expected effect of library sequence length.
Probability estimates calculated from the distribution of similarity scores are gener- ally conservative, as are probabilities calculated using the Altschul-Gish l, K, and H parameters. Thus, length-corrected similarity scores improve the sensitivity of data- base searches.
Sequence similarity searches today are the most The BLAST package of sequence comparison effective method for exploiting the information in programs Altschul et al. One of the most dramatic improvements in part because of its accurate statistical estimates. Gish, However, unrelated sequence database search. Several methods for estimating these par- empirical approach described here provides an ameters are outlined in Methods. These estimation methods have been lated-sequence similarity scores Figure 1.
For incorporated into versions 2. Distribution of sequence similarity z-scores. The number of sequences obtaining a similarity score z-score , calculated using the regress1 method, in x-axis bins of two z-score units A, B and C or four units C are shown. Symbols show the observed number of sequences; the continuous line indi- cates the expected distribution of z- scores for an extreme value distri- bution. The E - displayed in Figure 1.
For these examples, the expec- and N is the number of tests that have been per- tation values for the highest scoring unrelated formed. For similarity searches against a protein sequences ranged from 0.
The the local character of the sequence alignment. BLAST suite of programs also calculates an expec- Figure 1A and C also include examples where low tation, but typically reports the probability of gap penalties were used. Only selected related sequences are shown; all the highest-scoring unrelated sequences are shown.
The sequence length len and several similarity measures are shown. The z-sc column reports the length-corrected Z-score for the alignment. The E N value reports the number of times the score should be obtained by chance for a search against a data- base of size N. While high gap-penalties to 6.
Distribution of P - values, Smith-Waterman. The probabilities predicted frequency were sorted from lowest to highest and plotted as a fraction of the total number of searches 54, observed frequency. Distribution of P - values. With the lower gap-penalties, the sequence database see Methods.
The results for all 54 expected increase in unrelated sequence similarity sequences were combined by plotting the cumulat- score with length.
Figure 2. Thus, the regress1 and regress2 statistical with scores above the value; i. As before, we calculated the tein sequences when the Smith-Waterman z-value for the difference in performance between algorithm is used. Search performance, Smith-Waterman.
A, Performance using the equival- ence number criterion. The number of sequences performing better or worse left panel and the z-value of the difference right panel; Pearson, is shown. Search performance, PIR Search perform- ance using two sequences from each of 54 PIR39b families is shown.
A, Comparison Figure 5. Search performance, Smith-Waterman and using the equivalence number. When reference search per- forms better, the z-values are negative and indicate the our selection of the 54 query sequence families. Investigators often wonder: what P -value or E -value should be used to infer homology?
We have examined several strategies for correct- The answer to this question depends both on the ing the length-dependence of local protein number of searches that are being performed and sequence similarity scores. The default method the investigator's concern about inferring hom- used by programs in versions 2. In addition, Altschul-Gish scaling will have a homology assigned incorrectly. These authors recognized that the number databases.
His was done by Mott. In addition, it may be possible approach is similar to the regress2 estimation eval- to correct for the effect of hydrophobic patches and uated here, except that maximum likelihood esti- of low complexity regions. Since c Methods depends both on the amino acid composition of Sequence libraries and similarity searching the query sequence pu and each library sequence qv, it must be recalculated for each sequence com- Searches were performed on the annotated por- parison.
The approach requires additional sequence in the database has been assigned to a alignments to be calculated, and sequences with protein superfamily. The earlier experiments com- internal duplications can confuse the estimation pared the performance of two comparison procedure if the duplications are not recognized. This report focuses on the statistics since an arbitrary number of sub-optimal scores of high-scoring unrelated sequences, which can be generated from a pair of sequences.
In several cases the sequences and examining the distribution of for example, serine proteases, protein kinases, scores. These sequences from the sequences. Expectation values calculated from same superfamily were given the same superfam- database searches are often quite similar to those ily number.
Empirical Sequence Similarity Statistics 81 Searches were performed with 54 of the 67 Statistical estimates for scaled similarity scores query sequences selected from the PIR39 database listed in Table I of Pearson Thirteen of the Six methods, Altschul-Gish, log -scaled, previous superfamilies were excluded either scaled, unscaled, regress1, regress2, and because they were homologous with other superfa- regress3, were used to calculate statistical esti- milies immunoglobulin kappa V-I, kappa C, class- mates for similarity scores.
The estimation is straightforward if all the SwissProt34 query sets have 20 protein families in sequences are unrelated, as occurs when random common. Version 3. Similarity scores from related used. Estimates for regress1 and We also examined the performance of a simple regress3 regression-scaled scores are calculated by log-length correction log -scaled described by the following steps. Pearson
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