## Statistic Procedures – Hypothesis Tests for One Population Mean

We often use inferential statistics to make decision or judgments about the value of a parameter, such as a population mean. One of the most commonly used methods for making such decisions or judgements is to perform a hypothesis test. A hypothesis is a statement taht something is true. Typically, a hypothesis test involves two hypotheses: the null hypothesis and the alternative hypothesis (or research hypothesis), which we define as follows.

• Null hypothesis: A hypothesis is to be tested. We use the symbol H0 to represent the null hypothesis.
• Alternative hypothesis: A hypothesis to be considered as an alternative to the null hypothesis. We use the symbol Ha to represent the alternative hypothesis.
• Hypothesis test: The problem in a hypothesis test is to decide whether the null hypothesis should be rejected in favor of the alternative hypothesis.

The first step in setting up a hypothesis test is to decide on the null hypothesis and the alternative hypothesis. The following are some guidelines for choosing these two hypotheses. Although the guidelines refer specifically to hypothesis tests for one population mean, μ, they apply to any hypothesis test concerning one parameter.

Null hypothesis for a hypothesis test concerning a population mean, μ, always specifies a single value for that parameter. Hence we can express the null hypothesis as H0: μ = μ0, where μ0 is some number. The choice of the alternative hypothesis depdends on and should reflect the purpose of the hypothesis test. Three choices are possible for the alternative hypothesis: 1) If the primary concern is deciding whether a population mean, μ, is different from a specified value μ0, we express the alternative hypothesis as, Ha: μ != μ0, where a hypothesis test whose alternative hypothesis has this form is called a two-tailed test. 2) If the primary concern is deciding whether a population mean, μ, is less than a specified value μ0, we express the alternative hypothesis as, Ha: μ < μ0, where a hypothesis test whose alternative hypothesis has this form is called a left-tailed test. 3) If the primary concern is deciding whether a population mean, μ, is greater than a specified value μ0, we express the alternative hypothesis as, Ha: μ > μ0, where a hypothesis test whose alternative hypothesis has this form is called a right-tailed test. A hypothesis test is called a one-tailed test if it is either left tailed or right tailed. It is not uncommon that an sample mean falls within the area of acceptance for a two-tailed test but falls within the area of rejection for a one-tailed test. Therefore, a researcher who wishes to reject the null hypothesis may sometimes find that using a one-tailed rather a two-tailed test allows a previously nonsignificant result to become significant. For this reason, it is important that one-tailed test must depend on the nature of the hypothesis being tested and should therefore be decided at the outset of the research, rather than being decided afterward according to how the results turn out. One-tailed tests can only be used when there is a directional alternative hypothesis. This means that they may not be used unless results in only one direction are of interest and the possibility of the results being in the opposite direction is of no interest or consequence to the researcher.

PS: Results form Wiki

A two-tailed test is appropriate if the estimated value may be more than or less than the reference value, for example, whether a test taker may score above or below the historical average. A one-tailed test is appropirate if the estimated value may depart from the reference value in only one direction, for example, whether a machine produces more than one-percent defective products.

The basic logic hypothesis testing is that: Take a random sample from the population. If the sample data are consistent with the null hypothesis, do not reject the null hypothesis; if the sample data are inconsistent with the null hypothesis and supportive of the alternative hypothesis, reject the null hypothesis in favor of the alternative hypothesis. Suppose that a hypothesis test is conducted at a small significance level: If the null hypothesis is rejected, we conclude that the data provide sufficient evidence to support the alternative hypothesis. If the null hypothesis is not rejected, we conclude that the data do not provide sufficient evidence to support the alterantive hypothesis. Another way of viewing the use of a small significance level is as follows: The null hypothesis gets the benefit of the doubt; the alternative hypothesis has the burden of proof.

When the null hypothesis is rejected in a hypothesis test performed at the signifiance level α, we frequently express that fact with the phrase "the test results are statistically signifant at the α level." Simiarly, when the null hypothesis is not rejected in a hypothesis test performed at the sigificance level α, we often express that fact with the phrase "the test results are not statistically significant at the α level."

One-Mean z-Test (σ known)

The one-mean z-test is also known as the one-sample z-test and the one-variable z-test. We prefer "one-mean" because it makes clear the parameter being tested. Procedure 9.1 provides a step by step method for performing a one-mean z-test. As you can see, Procedure 9.1 includes options for either the critical-value approach or the P-value approach.

Properties and guidelines for use of the one-mean z-test are similar to those for the one-mean z-interval procedure. In particular, the one-mean z-test is robust to moderate violations of the normality assumption but, even for large samples, can sometimes be unduly affected by outlikers because the sample mean is not resistant to outliers. PS: By saying that the hypothesis test is exact, we mean that the true significance level equals α; by saying that it is approximately correct, we mean that the true significance level only approximately equals α.

One-Mean t-Test Type II Error

Hypothesis tests do not always yield correct conclusions; they have built-in margins of error. An important part of planning a study is to consider both types of errors that can be made and their effects. Recall that two types of errors are possible with hypothesis tests. One is a Type I error: rejecting a true null hypothesis. The other is a Type II error: not rejecting a false null hypothesis. Also recall that the probability of making a Type I error is called the significance level of the hypothesis test and is denoted α, and that the probability of making a Type II error is denoted β.

Computing Type II Error Probabilities

The probability of making a Type II error depends on the sample size, the significance level, and the true value of the parameter under consideration. Power Curve for a Oone-Mean z-Test

In modern statistical practice, analysts generally use the probability of not making a Type II error, called the power, to appraise the performance of a hypothesis test. Once we know the Type II error probability, β, obtaining the power is simple – we just substract β from 1. The power of a hypothesis test is between 0 and 1 and measures the ability of the hypothesis test to detect a false null hypothesis. If the power is near 0, the hypothesis test is not very good at detecting a false full hypothesis; if the power is near 1, the hypothesis test is extremely good at detect a false null hypothesis.

In reality, the true value of the parameter in question will be unkown. Consequently, construct a table of power for various values of the parameter consistent with the alternative hypothesis is helpful in evaluating the overall effectiveness of a hypothesis test. Even more helpful is a visual display of the effectiveness of the hypothesis test, obtained by plotting points of power against various values of the parameter and then connecting the points with a smooth curve. The resulting curve is called a power curve. In general, the closer a power is to 1, the better the hypothesis test is at detecting a false null hypothesis. Procedure 9.5 provides a step-by-step method for obtaining a power curve for a one-mean z-test. Sample Size and Power

For a fixed significance level, increasing the sample size increases the power. By using a sufficiently large sample size, we can obtain a hypothesis test with as much power as we want. However, in practice, larger sample sizes tend to increase the cost of a study. Consequently, we must balance, among other things, the cost of a large sample against the cost of possible errors. As we have indicated, power is a useful way to evaluate the overall effectiveness of a hypothesis-testing procedure. Additionally, power can be used to compare different procedures. For example, a researcher might decide between two hypothesis-testing procedures on the basis of which test is more powerful for the situation under consideration.

## Statistic Procedures – Confidence Interval Confidence Intervals for One Population Mean

A common problem in statistics is to obtain information about the mean, μ, of a population. One way to obtain information about a population mean μ without taking a census is to estimate it by a sample mean x(bar). So, a point estimate of a parameter is the value of a statistic used to estimate the parameter. More generally, a statistic is called an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, the statistic is called a biased estimator of the parameter. Ideally, we want our statistic to be unbiased and have small standard error. In that case, chances are good that our point estimate (the value of the statistic) will be close to the parameter.

However, it is not uncommon that a sample mean is usually not equal to the population mean, especially when the standard error is not small as stated previously. Therefore, we should accompany any point estimate of μ with information that indicates the accuracy of that estimate. This information is called a confidence-interval estimate for μ. By definition, the confidence interval (CI) is an interval of numbers obtain from a point estimate of a parameter. The confidence level is the confidence we have that the parameter lies in the confidence interval. And the confidence-interval estimate is the confidence level and confidence interval. An confidence interval for a population mean depends on the sample mean, x(bar), which in turn depdends on the sample selected.

Margin of error E indicates how accurate the sample mean of x(bar) is as an estimate for the value of the unknown parameter of μ. With the point estimate and confidence-interval estimate (of 95% confidence interval), we can be 95% confident that the μ is within E of the sample mean. Simply, it means that the μ = point estimate +- E.

Summary

• Point estimate
• Confidence-interval estimate
• Margin of error

Computing the Confidence-Interval for One Population Mean (σ known)

We not develop a step-by-step procedure to obtain a confidence interval for a population mean when the population standard deviation is known. In doing so, we assume that the variable under consideration is normallhy distributed. Because of the central limit theorem, however, the procedure will also work to obtain an approximately correct confidence interval when the sample size is large, regardless of the distribution of the variable. The basis of our confidence-interval procedure is the sampling distribution of the sample mean for a normally distributed variable: Suppose that a variable x of a population is normally distributed with mean μ and standard deviation σ. Then, for samples of size n, the variable x(bar) is also normally distributed and has mean μ and standard deviation σ/√n. As a consequence, we have the procedure to compute the confidence-interval. PS: The one-mean z-interval procedure is also known as the one-sample z-interval procedure and the one-variable z-interval procedure. We prefer "one-mean" because it makes clear the parameter being estimated.

PS: By saying that the confidence interval is exact, we mean that the true confidence level equals 1 – α; by saying that the confidence that the confidence interval is approximately correct, we mean that the true confidence level only approximately equals 1 – α.

Before applying Procedure 8.1, we need to make several comments about it and the assumptions for its use, including:

• We use the term normal population as an abbreviation for "the variable under consideration is normally distributed."
• The z-interval procedure works reasonably well even when the variable is not normally distributed and the sample size is small or moderate, provided the variable is not too far from being normally distributed. Thus we say that the z-interval procedure is robust to moderate violations of the normality assumption.
• Watch for outlilers because their presence calls into question the normality assumption. Moreover, even for large samples, outliers can sometimes unduly affect a z-interval because the sample mean is not resistant to outliers.
• A statistical procedure that works reasonably well even when one of its assumptions is violated (or moderately violated) is called a robust procedure relative to that assumption.

Summary Key Fact 8.1 makes it clear that you should conduct preliminary data analyses before applying the z-interval procedure. More generally, the following fundamental principle of data analysis is relevant to all inferential procedures: Before performing a statistical-inference procedure, examine the sample data. If any of the conditions required for using the procedure appear to be violated, do not apply the procedure. Instead use a different, more appropriate procedure, if one exists. Even for small samples, where graphical displays must be interpreted carefully, it is far better to examine the data than not to. Remember, though, to proceed cautiously when conducting graphical analyses of small samples, especially very small samples – say, of size 10 or less.

Sample Size Estimation

If the margin of error and confidence level are specified in advance, then we must determine the sample size needed to meet those specifications. To find the formula for the required sample, we solve the margin-of-error formula, E = zα/2 · σ/√n, for n. See the computing formula in Formula 8.2. Computing the Confidence-Interval for One Population Mean (σ unknown)

So far, we have discussed how to obtain the confidence-interval estimate when the population standard deviation, σ, is known. What if, as is usual in practice, the population standard deviation is unknown? Then we cannot base our confidence-interval procedure on the standardized version of x(bar). The best we can do is estimate the population standard deviation, σ, by the sample standard deviation, s; in other words, we replace σ by s in Procedure 8.1 and base our confidence-interval procedure on the resulting variable t (studentized version of x(bar)). Unlike the standardize version, the studentized version of x(bar) does not have a normal distribution.

Suppose that a variable x of population is normally distributed with mean μ. Then, for samples of size n, the variable t has the t-distribution with n-1 degrees of freedom. A variable with a t-distribution has an associated curve, called a t-curve. Although there is a different t-curve for each number of degrees of freedom, all t-curves are similar and resemble the standard normal cruve. As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve.

Having discussed t-distributions and t-curves, we can now develop a procedure for obtaining a confidence interval for a population mean when the population standard deviation is unknown. The procedure is called the one-mean t-interval procedure or, when no confusion can arise, simply the t-interval procedure. Properties and guidelines for use of the t-interval procedure are the same as those for the z-interval procedure. In particular, the t-interval procedure is robust to moderate violations of the normality assumption but, even for large samples, can sometimes be unduly affected by outliers because the sample mean and sample standard deviation are not resistant to outliers.

What If the Assumptions Are Not Satisfied?

Suppose you want to obtain a confidence interval for a population mean based on a small sample, but preliminary data analyses indicate either the presence of outliers or that the variable under consideration is far from normally distributed. As neither the z-interval procedure nor the t-interval procedure is appropriate, what can you do? Under certain conditions, you can use a nonparametric method. Most nonparametric methods do not require even approximate normality, are resistant to outliers and other extreme values, and can be applied regardless of sample size. However, parametric methods, such as the z-interval and t-interval procedures, tend to give more accurate results than nonparametric methods when the normality assumption and other requirements for their use are met.

## Inherited Variation and Polymorphism in DNA

The original Human Genome Project and the subsequent study of now many thousands of individuals worldwide have provided a vast amount of DNA sequence information. With this information in hand, one can begin to characterize the types and frequencies of polymorphic variation found in the human genome and to generate catalogues of human DNA sequence diversity around the globe. DNA polymorphisms can be classified according to how the DNA sequence varies between the different alleles. #### Single Nucleotide Polymorphisms

The simplest and most common of all polymorphisms are single nucleotide polymorphisms (SNPs). A locus characterized by a SNP usually has only two alleles, corresponding to the two different bases occupying that particular location in the genome. As mentioned previously, SNPs are common and are observed on average once every 1000 bp in the genome. However, the distribution of SNPs is uneven around the genome; many more SNPs are found in noncoding parts of the genome, in introns and in sequences that are some distance from known genes. Nonetheless, there is still a significant number of SNPs that do occur in genes and other known functional elements in the genome. For the set of protein-coding genes, over 100,000 exonic SNPs have been documented to date. Approximately half of these do not alter the predicted amino acid sequence of the encoded protein and are thus termed synonymous, whereas the other half do alter the amino acid sequence and are said to be nonsynonymous. Other SNPs introduce or change a stop codon, and yet others alter a known splice site; such SNPs are candidates to have significant functional consequences.

The significance for health of the vast majority of SNPs is unknown and is the subject of ongoing research. The fact that SNPs are common does not mean that they are without effect on health or longevity. What it does mean is that any effect of common SNPs is likely to involve a relatively subtle altering of disease susceptibility rather than a direct cause of serious illness.

#### Insertion-Deletion Polymorphisms

A second class of polymorphism is the result of variations caused by insertion or deletion (in/dels or simply indels) of anywhere from a single base pair up to approximately 1000 bp, although larger indels have been documented as well. Over a million indels have been described, numbering in the hundreds of thousands in any one individual’s genome. Approximately half of all indels are referred to as “simple” because they have only two alleles – that is, the presence or absence of the inserted or deleted segment.

#### Microsatellite Polymorphisms

Other indels, however, are multiallelic due to variable numbers of the segment of DNA that is inserted in tandem at a particular location, thereby constituting what is referred to as a microsatellite. They consist of stretches of DNA composed of units of two, three, or four nucleotides, such as TGTGTG, CAACAACAA, or AAATAAATAAAT, repeated between one and a few dozen times at a particular site in the genome. The different alleles in a microsatellite polymorphism are the result of differing numbers of repeated nucleotide units contained within any one microsatellite and are therefore sometimes also referred to as short tandem repeat (STR) polymorphisms. A microsatellite locus often has many alleles (repeat lengths) that can be rapidly evaluated by standard laboratory procedures to distinguish different individuals and to infer familial relationships. Many tens of thousands of microsatellite polymorphic loci are known throughout the human genome. Finally, microsatellites are a particularly useful group of indels. Determining the alleles at multiple microsatellite loci is currently the method of choice for DNA fingerprinting used for identity testing.

#### Mobile Element Insertion Polymorphisms

Nearly half of the human genome consists of families of repetitive elements that are dispersed around the genome. Although most of the copies of these repeats are stationary, some of them are mobile and contribute to human genetic diversity through the process of retrotransposition, a process that involves transcription into an RNA, reverse transcription into a DNA sequence, and insertion into another site in the genome. Mobile element polymorphisms are found in nongenic regions of the genome, a small proportion of them are found within genes. At least 5000 of these polymorphic loci have an insertion frequency of greater than 10% in various populations.

#### Coyp Number Variants

Another important type of human polymorphism includes copy number variants (CNVs). CNVs are conceptually related to indels and microsatellites but consist of variation in the number of copies of larger segments of the genome, ranging in size from 1000 bp to many hundreds of kilobase pairs. Variants larger than 500 kb are found in 5% to 10% of individuals in the general population, whereas variants encompassing more than 1 Mb are found in 1% to 2%. The largest CNVs are sometimes found in regions of the genome characterized by repeated blocks of homologous sequences called segmental duplications (or segdups).

Smaller CNVs in particular may have only two alleles (i.e., the presence or absence of a segment), similar to indels in that regard. Larger CNVs tend to have multiple alleles due to the presence of different numbers of copies of a segment of DNA in tandem. In terms of genome diversity between individuals, the amount of DNA involved in CNVs vastly exceeds the amount that differs because of SNPs. The content of any two human genomes can differ by as much as 50 to 100 Mb because of copy number differences at CNV loci.

Notably, the variable segment at many CNV loci can include one to as several dozen genes, that thusCNVs are frequently implicated in traits that involve altered gene dosage. When a CNV is frequent enough to be polymorphic, it represents a background of common variation that must be understood if alterations in copy number observed in patients are to be interpreted properly. As with all DNA polymorphism, the significance of different CNV alleles in health and disease susceptibility is the subject of intensive investigation.

#### Inversion Polymorphisms

A final group of polymorphisms to be discussed is inversions, which differ in size from a few base pairs to large regions of the genome (up to several megabase pairs) that can be present in either of two orientations in the genomes of different individuals. Most inversions are characterized by regions of sequence homology at the edges of the inverted segment, implicating a process of homologous recombination in the origin of the inversions. In their balanced form, inversions, regardless of orientation, do not involve a gain or loss of DNA, and the inversion polymorphisms (with two alleles corresponding to the two orientations) can achieve substantial frequencies in the general population.