Strictly speaking a 95% confidence interval means that if we were to take 100 different samples and compute a 95% confidence interval for each sample, then approximately 95 of the 100 confidence intervals will contain the true mean value (μ). Suppose we want to generate a 95% confidence interval estimate for an unknown population mean. Using algebra, we can rework this inequality such that the mean (μ) is the middle term, as shown below.

In the health-related publications a 95% confidence interval is most often used, but this is an arbitrary value, and other confidence levels can be selected. With smaller samples (n< 30) the Central Limit Theorem does not apply, and another distribution called the t distribution must be used.

Suppose we wish to estimate the mean systolic blood pressure, body mass index, total cholesterol level or white blood cell count in a single target population.

Descriptive statistics on variables measured in a sample of a n=3,539 participants attending the 7th examination of the offspring in the Framingham Heart Study are shown below.

The table below shows data on a subsample of n=10 participants in the 7th examination of the Framingham offspring Study.

Suppose we compute a 95% confidence interval for the true systolic blood pressure using data in the subsample. Suppose we wish to estimate the proportion of people with diabetes in a population or the proportion of people with hypertension or obesity. This formula is appropriate for large samples, defined as at least 5 successes and at least 5 failures in the sample. Specific applications of estimation for a single population with a dichotomous outcome involve estimating prevalence, cumulative incidence, and incidence rates. The table below, from the 5th examination of the Framingham Offspring cohort, shows the number of men and women found with or without cardiovascular disease (CVD).

There are many situations where it is of interest to compare two groups with respect to their mean scores on a continuous outcome. We could begin by computing the sample sizes (n1 and n2), means 1 and 2), and standard deviations (s1 and s2) in each sample.

The confidence interval will be computed using either the Z or t distribution for the selected confidence level and the standard error of the point estimate.

Where Sp is the pooled estimate of the common standard deviation (assuming that the variances in the populations are similar) computed as the weighted average of the standard deviations in the samples. Suppose we want to compare mean systolic blood pressures in men versus women using a 95% confidence interval. The following table includes 95% confidence intervals for each characteristic, computed using the same formula we used for the confidence interval for the difference in mean systolic blood pressures. The confidence interval for the difference in means provides an estimate of the absolute difference in means of the outcome variable of interest between the comparison groups. We previously considered a subsample of n=10 participants attending the 7th examination of the Offspring cohort in the Framingham Heart Study. Suppose we wish to construct a 95% confidence interval for the difference in mean systolic blood pressures between men and women using these data.

The previous section dealt with confidence intervals for the difference in means between two independent groups. A single sample of participants and each participant is measured twice, once before and then after an intervention. A goal of these studies might be to compare the mean scores measured before and after the intervention, or to compare the mean scores obtained with the two conditions in a crossover study.

For example, we might be interested in the difference in an outcome between twins or between siblings. In the one sample and two independent samples applications participants are the units of analysis.

However, with two dependent samples application,the pair is the unit (and not the number of measurements which is twice the number of units).

In the Framingham Offspring Study, participants attend clinical examinations approximately every four years. We can now use these descriptive statistics to compute a 95% confidence interval for the mean difference in systolic blood pressures in the population. A crossover trial is conducted to evaluate the effectiveness of a new drug designed to reduce symptoms of depression in adults over 65 years of age following a stroke. One can compute a risk difference, which is computed by taking the difference in proportions between comparison groups and is similar to the estimate of the difference in means for a continuous outcome.

The risk ratio (or relative risk) is another useful measure to compare proportions between two independent populations and it is computed by taking the ratio of proportions.

Note that this formula is appropriate for large samples (at least 5 successes and at least 5 failures in each sample).

The following table contains data on prevalent cardiovascular disease (CVD) among participants who were currently non-smokers and those who were current smokers at the time of the fifth examination in the Framingham Offspring Study.

A randomized trial is conducted among 100 subjects to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery.

Using the data in the table below, compute the point estimate for the difference in proportion of pain relief of 3+ points.are observed in the trial. Compute the 95% confidence interval for the difference in proportions of patients reporting relief (in this case a risk difference, since it is a difference in cumulative incidence).

The risk difference quantifies the absolute difference in risk or prevalence, whereas the relative risk is, as the name indicates, a relative measure.

The relative risk is a ratio and does not follow a normal distribution, regardless of the sample sizes in the comparison groups.

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The surgical registrar who investigated appendicitis cases, referred to in Chapter 3 , wonders whether the percentages of men and women in the sample differ from the percentages of all the other men and women aged 65 and over admitted to the surgical wards during the same period. The calculation of the standard error of a difference in proportions p1 - p2 follows the same logic as the calculation of the standard error of two means; sum the squares of the individual standard errors and then take the square root. Note that this is an approximate formula; the exact one would use the population proportions rather than the sample estimates.

For a significance test we have to use a slightly different formula, based on the null hypothesis that both samples have a common population proportion, estimated by p. To obtain p we must amalgamate the two samples and calculate the percentage of women in the two combined; 100 - p is then the percentage of men in the two combined. It is important to note that the deaths must be independently caused; for example, they must not be the result of an epidemic such as influenza. This method should be regarded as giving no more than approximate but useful guidance, and is unlikely to be valid over a period of more than very few years owing to changes in diagnostic techniques. Sometimes it is possible to record the results of treatment or some sort of test or investigation as one of two alternatives. The significance of the results can then be simply tested by McNemar's test in the following way.

However, in practice, the fairly small numbers that form the subject of this type of investigation make a correction advisable.

For example, a registrar in the gastroenterological unit of a large hospital in an industrial city sees a considerable number of patients with severe recurrent aphthous ulcer of the mouth. Over a period of 6 months the registrar selected every patient with this disorder and paired them off as far as possible by reference to age, sex, and frequency of ulceration.

Although this does not include zero, the confidence interval is quite wide, reflecting uncertainty as to the true difference because the sample size is small. Why is the standard error used for calculating a confidence interval for the difference in two proportions different from the standard error used for calculating the significance? It is worth pointing out that the formula for calculating the standard error of an estimate is not necessarily unique: it depends on underlying assumptions, and so different assumptions or study designs will lead to different estimates for standard errors for data sets that might be numerically identical. A dermatologist tested a new topical application for the treatment of psoriasis on 47 patients. Get the insider's tips and tricks in how to survive your math course and ace the next test. A proportion p of the confidence intervals would contain the population parameter in question.

Several example problems with step by step illustrations of calculating confidence intervals are presented.

Confidence intervals for the difference between two population proportions, and two population means are given for detailed explanation on how to calculate confidence intervals.

Concept map showing inter-connections of new concepts in this tutorial and those previously introduced.

After the students complete this course, they should be able to use the statistical methods to analyze certain data and interpret the results in a practical meaning. Evaluate the difference between two population means and proportions by constructing a confidence interval and conducting hypothesis test. Evaluate the difference between more than two population means by performing an analysis of variance (ANOVA) and multiple comparisons of means. Technical Requirements & SkillsOne of the greatest barriers to taking an online course is a lack of basic computer literacy. Students can use statistical programs such as SPSS or Minitab; however, this is not required. Accessibility And AccommodationPlease visit our ADA Compliance webpage for information about accessibility involving the tools used in this course. Proctored Exam PolicyThrough a careful examination of this syllabus, it is the student’s responsibility to determine whether this online course requires proctored exams. Expectations Of This CourseThis is an online course (with an on-campus final exam), meaning most of the course work will be conducted online. Messages is a private and secure text-based communication system which occurs within a course among its Course members. Visit our Writing Resources webpage for more information on professional writing and technical communication skills. Discussion ForumsThe discussion forum will be used for class discussions and other postings of general interest that are directly related to the course, however there are no requirements for discussion. When you submit your homework and exams, your scores will be released to you automatically. Weekly learning objective, class notes, assignments and important dates are in course content. Many of the outcomes we are interested in estimating are either continuous or dichotomous variables, although there are other types which are discussed in a later module. In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean. This means that there is a 95% probability that the confidence interval will contain the true population mean. However, if the sample size is large (n > 30), then the sample standard deviations can be used to estimate the population standard deviation. Note that for a given sample, the 99% confidence interval would be wider than the 95% confidence interval, because it allows one to be more confident that the unknown population parameter is contained within the interval. The t distribution is similar to the standard normal distribution but takes a slightly different shape depending on the sample size. We select a sample and compute descriptive statistics including the sample size (n), the sample mean, and the sample standard deviation (s). The margin of error is very small (the confidence interval is narrow), because the sample size is large.

Because the sample size is small, we must now use the confidence interval formula that involves t rather than Z. These diagnoses are defined by specific levels of laboratory tests and measurements of blood pressure and body mass index, respectively. For example, we might be interested in comparing mean systolic blood pressure in men and women, or perhaps compare BMI in smokers and non-smokers.

For analysis, we have samples from each of the comparison populations, and if the sample variances are similar, then the assumption about variability in the populations is reasonable.

The sample is large (> 30 for both men and women), so we can use the confidence interval formula with Z. Next we substitute the Z score for 95% confidence, Sp=19, the sample means, and the sample sizes into the equation for the confidence interval. The following table contains descriptive statistics on the same continuous characteristics in the subsample stratified by sex. There is an alternative study design in which two comparison groups are dependent, matched or paired. Participants are usually randomly assigned to receive their first treatment and then the other treatment. Symptoms of depression are measured on a scale of 0-100 with higher scores indicative of more frequent and severe symptoms of depression.

When the outcome is dichotomous, the analysis involves comparing the proportions of successes between the two groups. The risk ratio is a good measure of the strength of an effect, while the risk difference is a better measure of the public health impact, because it compares the difference in absolute risk and, therefore provides an indication of how many people might benefit from an intervention. When constructing confidence intervals for the risk difference, the convention is to call the exposed or treated group 1 and the unexposed or untreated group 2. However, the natural log (Ln) of the sample RR, is approximately normally distributed and is used to produce the confidence interval for the relative risk.

After excluding his sample of appendicitis cases, so that they are not counted twice, he makes a rough estimate of the number of patients admitted in those 10 years and finds it to be about 12-13 000. For Normally distributed variables these two are the same if the standard deviations are assumed to be the same, but in the binomial case the standard deviations depend on the estimates of the proportions, and so if these are different so are the standard deviations. It is based on the alternative hypothesis that there is a real difference in proportions (further discussion on this point is given in Common questions at the end of this chapter). If the population of the town or area where they occur is fairly large, say, some thousands, and provided that the deaths are independent of one another, the standard error of the number of deaths from a specified cause is given approximately by its square root, Further, the standard error of the difference between two numbers of deaths, and , can be taken as . The reports of the deaths must likewise be independent; for example, the criteria for diagnosis must be consistent from year to year and not suddenly change in accordance with a new fashion or test, and the population at risk must be the same size over the period of study. For instance, in Carlisle the number of deaths from ischaemic heart disease in 1973 was 276.

For instance, two treatments or tests might be carried out on pairs obtained by matching individuals chosen by random sampling, or the pairs might consist of successive treatments of the same individual (see Chapter 7 for a comparison of pairs by the tt test). Claims have been made that a recently introduced preparation stops the pain of these ulcers and promotes quicker healing than existing preparations. Therefore we may conclude that treatment A gave significantly better results than treatment B.

If we suppose that a nominal variable simply takes the value 0 or 1, then the mean is simply the proportion of is and the standard deviation is directly dependent on the mean, being largest when the mean is 0.5. He applied it to the lesions on one part of the patient's body and what he considered to be the best traditional remedy to the lesions on another but comparable part of the body, the choice of area being made by the toss of a coin. Take an SRS of size n from a large population that contains a proportion parameter p of successes. In statistics, interval estimation is the use of sample data to calculate an interval of possible (or probable) values of an unknown population parameter. This course will provide the students with a fundamental tool for their future major course study and research.

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By computer literacy we mean being able to manage and organize computer files efficiently, and learning to use your computer's operating system and software quickly and easily. Please visit our Proctored Exam Resources webpage for important information concerning proctored exams, proctoring centers, and important forms. The parameters to be estimated depend not only on whether the endpoint is continuous or dichotomous, but also on the number of groups being studied. Recall that sample means and sample proportions are unbiased estimates of the corresponding population parameters. In a sense, one could think of the t distribution as a family of distributions for smaller samples.

The formulas for confidence intervals for the population mean depend on the sample size and are given below. Both of these situations involve comparisons between two independent groups, meaning that there are different people in the groups being compared. The standard error of the point estimate will incorporate the variability in the outcome of interest in each of the comparison groups.

The men have higher mean values on each of the other characteristics considered (indicated by the positive confidence intervals).

This judgment is based on whether the observed difference is beyond that expected by chance. We compute the sample size (which in this case is the number of distinct participants or distinct pairs), the mean and standard deviation of the difference scores, and we denote these summary statistics as n, d and sd, respectively.

The data below are systolic blood pressures measured at the sixth and seventh examinations in a subsample of n=15 randomly selected participants. This is similar to a one sample problem with a continuous outcome except that we are now using the difference scores. Because the 95% confidence interval for the mean difference does not include zero, we can conclude that there is a statistically significant difference (in this case a significant improvement) in depressive symptom scores after taking the new drug as compared to placebo.

Here smoking status defines the comparison groups, and we will call the current smokers group 1 and the non-smokers group 2.

Because the 95% confidence interval includes zero, we conclude that the difference in prevalent CVD between smokers and non-smokers is not statistically significant.

Patients are randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery. Both of these situations involve comparisons between two In contrast, dependent groups are matched or paired in some way. The result might then be recorded as "responded or did not respond", "improved or did not improve", "positive or negative", and so on. Let the larger number of pairs in either of rows (2) or (3) be called n1 and the smaller number of pairs in either of those two rows be n2. To one member of each pair, chosen by the toss of a coin, she gave treatment A, which she and her colleagues in the unit had hitherto regarded as the best; to the other member she gave the new treatment, B. The null and alternative hypotheses are hypotheses about means, either that they are the same (null) or different (alternative). It covers making inference based on two samples, analysis of variance, simple linear regression, categorical data analysis and nonparametric statistics. Keep in mind that this is not a computer literacy course; but students enrolled in online courses are expected to have moderate proficiency using a computer. In fact, online courses require a degree of self-motivation, self-discipline, and technology skills which can make these courses more demanding for some students. When the outcome of interest is dichotomous like this, the record for each member of the sample indicates having the condition or characteristic of interest or not. If we assume equal variances between groups, we can pool the information on variability (sample variances) to generate an estimate of the population variability. However,we will first check whether the assumption of equality of population variances is reasonable. The appropriate formula for the confidence interval for the mean difference depends on the sample size.

Since the data in the two samples (examination 6 and 7) are matched, we compute difference scores by subtracting the blood pressure measured at examination 7 from that measured at examination 6 or vice versa. The trial was run as a crossover trial in which each patient received both the new drug and a placebo. It is the ratio of the odds or disease in those with a risk factor compared to the odds of disease in those without the risk factor. A confidence interval for the difference in prevalent CVD (or prevalence difference) between smokers and non-smokers is given below.

By convention we typically regard the unexposed (or least exposed) group as the comparison group, and the proportion of successes or the risk for the unexposed comparison group is the denominator for the ratio. First, a confidence interval is generated for Ln(RR), and then the antilog of the upper and lower limits of the confidence interval for Ln(RR) are computed to give the upper and lower limits of the confidence interval for the RR.

Note that this test gives results identical to those obtained by the test without continuity correction (described in Chapter 7). Thus for nominal variables the standard deviations (and thus the standard errors) will also be different for the null and alternative hypotheses. What is the standard error of the difference between the percentages at this hospital and the first? The sample should be representative of the population, with participants selected at random from the population. In contrast, when comparing two independent samples in this fashion the confidence interval provides a range of values for the difference. If a 95% confidence interval includes the null value, then there is no statistically meaningful or statistically significant difference between the groups. Based on this interval, we also conclude that there is no statistically significant difference in mean systolic blood pressures between men and women, because the 95% confidence interval includes the null value, zero. When the samples are dependent, we cannot use the techniques in the previous section to compare means.

Before receiving the assigned treatment, patients are asked to rate their pain on a scale of 0-10 with high scores indicative of more pain.

This is approximately Normally distributed under the null hypothesis, and its probability can be read from appendix-table.pdftable A. Consequently to avoid bias in the assessment of the results a colleague recorded the results of treatment without knowing which patient in each pair had which treatment. For a confidence interval, the alternative hypothesis is assumed to be true, whereas for a significance test the null hypothesis is assumed to be true. What is the difference between these percentages of forceps delivery with a 95% confidence interval and what is its significance? In generating estimates, it is also important to quantify the precision of estimates from different samples. Rather, it reflects the amount of random error in the sample and provides a range of values that are likely to include the unknown parameter. Just as with large samples, the t distribution assumes that the outcome of interest is approximately normally distributed. If the confidence interval does not include the null value, then we conclude that there is a statistically significant difference between the groups.

Because the samples are dependent, statistical techniques that account for the dependency must be used.

Each patient is then given the assigned treatment and after 30 minutes is again asked to rate their pain on the same scale.

In general the difference in the values of the two methods of calculating the standard errors is likely to be small, and use of either would lead to the same inferences. Another way of thinking about a confidence interval is that it is the range of likely values of the parameter (defined as the point estimate + margin of error) with a specified level of confidence (which is similar to a probability). For each of the characteristics in the table above there is a statistically significant difference in means between men and women, because none of the confidence intervals include the null value, zero.

Since the interval contains zero (no difference), we do not have sufficient evidence to conclude that there is a difference. When the outcome is continuous, the assessment of a treatment effect in a crossover trial is performed using the techniques described here. The difference in depressive symptoms was measured in each patient by subtracting the depressive symptom score after taking the placebo from the depressive symptom score after taking the new drug. All of these measures (risk difference, risk ratio, odds ratio) are used as measures of association by epidemiologists, and these three measures are considered in more detail in the module on Measures of Association in the core course in epidemiology. This is clearly much less than 1.96 times the standard error at the 5% level of probability. The reason this is mentioned here is that there is a close connection between the test of significance described in this chapter and the Chi square test described in Chapter 8.

Note also that this 95% confidence interval for the difference in mean blood pressures is much wider here than the one based on the full sample derived in the previous example, because the very small sample size produces a very imprecise estimate of the difference in mean systolic blood pressures. Confidence interval estimates for the risk difference, the relative risk and the odds ratio are described below. The difference in the arithmetic for the significance test, and that for calculating the confidence interval, could lead some readers to believe that they are unrelated, whereas in fact they are complementary. The problem does not arise with continuous variables, where the standard deviation is usually assumed independent of the mean, and is also assumed to be the same value under both the null and alternative hypotheses.

When there are small differences between groups, it may be possible to demonstrate that the differences are statistically significant if the sample size is sufficiently large, as it is in this example.

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