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20.03.2015
Mark has done a great job in writing complex statistical concepts in an easy to understand format that makes grasping them both easy to understand and to use. With the help of Mark's book, and some diligent studying, I received an A in my stats course. Excel Statistical Master that taught me all of the basic concepts for the different tests we used.
Unlike the indecipherable jargon in the countless books I have wasted money on, the language in this book is plain and easy to understand.
This same problem above is solved in the Excel Statistical Master with only 3 Excel formulas (and NO looking anything up on a Z Chart). This same problem above is solved in the Excel Statistical Master with only 3 Excel formulas (and not having to look anything up on a Z Chart). This same problem above is solved in the Excel Statistical Master with only 1 Quick Excel formula (and NO looking anything up on a Z Chart). This same problem above is solved in the Excel Statistical Master with only 2 Excel formulas (and not having to look anything up on a Z Chart). This would state that the actual population Proportion has a 95%probability of lying within the calculated interval. Statistics Confidence level is used to establish the statistics confidence interval it makes easily reached of the given data.
A confidence interval is a range of values that describes the uncertainty surrounding an estimate. One-sided confidence interval bound or limit since either the upper limit will be infinity or the lower limit will be minus infinity depending on whether it is a lower bound or an upper bound respectively. Calculating Confidence Intervals is a very importat operation within Confidence Intervals study. The confidence interval is a range of values that has a given probability of containing the true value of the association. Confidence limits for the mean are an interval estimate for the mean, whereas the estimate of the mean varies from sample to sample.
A confidence interval is an interval estimate of a population parameter and is used to indicate the reliability of an estimate. As noted in earlier modules a key goal in applied biostatistics is to make inferences about unknown population parameters based on sample statistics. 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. You are asked to evaluate a gauge used to measure the height of a certain aerospace component.
Bias is the difference between the measurement’s average performed by an operator (MA) and a reference value (RV) obtained by using a master gauge. It’s the difference in bias between a master gauge and the observed average over the complete operating range of the testing gauge. A sample of seven aerospace components of the same type was selected to cover the process’ operating range (variation).
The coefficient of determination R2(R-square) represents the fraction of variation in y explained by x (between 0 and 1). A sample of five nails of the same type was selected to cover the process’ operating range.
Since “0” is included in the CI, the hypothesis that the regression line passes through the origin (0,0) isn’t rejected. Variation among repeated measurements performed by one operator to a single unit and with the same measurement instrument.
An alternate method to the classic Gauge R&R is the Analysis of variance or ANOVA (see Montgomery and Runger, 1993, and MSA, 2002).

The advantages of the ANOVA are: a) The variances can be estimated with greater precision, and b) More information is obtained such as the interaction between the units and the operators. By comparing ANOVA’s F-values versus F-table values (see F-Table), it is observed that the statistically significant factors are the units, and the interaction between them and the operators. According to this comparison, the classic GR&R is underestimating the variation because it doesn’t include the interaction between units and operators.
Take a minimum of three operators and thirty units or events, some of which slightly outside of their specification limits. The AAA will be applied to a measurement system to evaluate a certain characteristic using a binary scale. Uncertainty of measurement is defined as “a parameter associated with the result of a measurement that characterizes the dispersion of the values that could reasonably be attributed to the measurand” (VIM 1993; GUM 1993). See also the Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results.
It’s obtained for instance from a manufacturer’s handbook or from a calibration certificate (outside sources) or from an assumed distribution. TUR: Test Uncertainty Ratio is the ratio of the accuracy tolerance of the unit under calibration to the accuracy tolerance of the calibration standard used.
TAR: Test Accuracy Ratio is the ratio of the accuracy tolerance of the unit under calibration to the uncertainty of the calibration standard used. The values in the denominator may be considered an approximation to the corresponding standard deviation $$\pm 3 \sigma$$.
It is customary to include a coverage (or security) factor, k, to the combined uncertainty to define a range of values in which the measurand will lie with a certain confidence level. To assess if the expanded uncertainty is acceptable, a rule-of-thumb is that the ratio of the tolerance of the measured unit to the maximum Ue be at least 4. I purchased the materials about halfway through the course and wish I had known about this manual from the start of the class! You'll be able to grasp your statistics course a LOT easier with the Excel Statistical Master. If you found your statistics book confusing, You'll really like the Excel Statistical Master.
Basically, it is concerned with the collection, organization, interpretation, calculation, forecast and analysis of statistical data.
All confidence intervals include zero, the lower bounds are negative whereas the upper bounds are positive. A 95% confidence interval provides a range of likely values for the parameter such that the parameter is included in the interval 95% of the time in the long term. If we careful, constructing a one-sided confidence interval can provide a more effective statement than use of a two-sided interval would. This section will help you to get knowledge over this.Confidence interval is defined as the function of manipulative of sample mean subtracted by error enclosed for the population mean of the function Formula for measuring the confidence interval by using the function is given by, confidence interval = $\bar{x} - EBM$ Error bounded for the sample mean is defined as the computation of tscore value for the confidence interval which is multiplied to the standard deviation value divided by the total values given in the data set. Confidence intervals are preferable to P-values, as they tell us the range of possible effect sizes compatible with the data. The interval estimate gives an indication of how much uncertainty in estimate of the true mean. The confidence intervals should overlap completely with each other if we take two samples from the same population.
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. Without assessing stability it's not possible to assure reliable evaluation of the other statistical properties. One of such components is selected and using the master gauge [A measuring device of a standard size that is used to calibrate other measuring instruments] its height was measured giving 2.48 cms.
It’s the change in bias with respect to the units’ variation of the characteristic being measured. Take at least 5 units (g>=5) that cover the process operating range and measure them with the master gauge.
Fit a regression line [a smooth curve fitted to the set of paired data in regression analysis; for linear regression the curve is a straight line] y=b0+b1x, where b0=intersection with the y-axis and b1=slope. One operator randomly measured its height 10 times each with the testing gauge, and once with the master gauge as the reference value (see table). One operator randomly measured its length 10 times each with the testing gauge, and once with the master gauge as the reference value (see table). This means that there exists a statistically significant difference between the sampled nails and the interaction is also important. In the case where the interaction effect is not statistically significant, those terms can be eliminated and the new computed F-values are obtained by dividing the corresponding MS by MSE.
A specific case is based on the binary scale: defective, non-defective, success or failure, etc.
Uncertainty is an important factor of the measurement process, ranging throughout all steps involved in metrology.
ISO Technical Advisory Group and the International Committee on Weights and Measures (CIPM) (corrected and reprinted in 1995). The positive square root of the number, which may be considered an approximation to the corresponding variance and is obtained from an assumed probability distribution.
While dealing with a large data, the confidence interval is a very important concept in statistics. A 95% confidence interval does'nt mean that particular interval has 95% chance of capturing the actual value of the parameter.
P-values simply provide a cut-off beyond which we assert that the findings are ‘statistically significant’.
In this case, the standard deviation is replaced by the estimated standard deviation 's', also known as the standard error.
If the confidence intervals overlap, the two estimates are deemed to be not significantly different. An error bar can be represent the standard deviation, but more often than not it shows the 95% confidence interval of the mean.
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. Next, 100 measurements were performed by a single operator during 4 weeks, once a day and 5 measurements each day (20 subgroups of size 5). Uncertainty leaves room for analysts to prove or disprove the results that are currently believed to be true rather than accepted the data as unbiased facts.
This can calculated by divided the accuracy tolerance from the assumed distribution by the accuracy tolerance of the selected standard.
This can calculated by divided the accuracy tolerance from the assumed distribution by the uncertainty of the selected standard. A confidence interval represents the long term chances of capturing the actual value of the population parameter over many different samples. With the analytic results, we can determining the P value to see whether the result is statistically significant. Since the standard error is an estimate for the true value of the standard deviation, the distribution of the sample mean $\bar x$ is no longer normal with mean $\mu$ and standard deviation $\frac{\sigma}{\sqrt{n}}$. However, this is a conservative test of significance that is appropriate when reporting multiple comparisons but the rates may still be significantly different at the 0.05 significance level even if the confidence intervals overlap.
The 95% confidence interval is an interval constructed such that in 95% of samples the true value of the population mean will fall within its limits. 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.
The additional information concerning about the point is conventional in the given data set.
Confidence interval for the population mean, based on a simple random sample of size n, for population with unknown mean and unknown standard deviation, is$\bar x$ ± t $\frac{s}{\sqrt{n}}$where, t is the upper $\frac{1-C}{2}$ critical value for the t distribution with n - 1 degrees of freedom, t(n - 1).
When comparing two parameter estimates, it is always true that if the confidence intervals do not overlap, then the statistics will be statistically significantly different.
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.
Confidence intervals deliberated for determining the means that are the intervals construct by means of a method that they will enclose to the population mean of a particular segment of the time period, on average is moreover 95% or 99% of the confidence level. Understanding that relation requires the concept of no effect, that is no difference between the groups being compared. This is why error bar showing 95% confidence interval are so useful on graphs, because if the bars of any two means do not overlap then we can infer that these mean are from different populations.
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.
For effect sizes that involve subtraction and for slopes and correlations, no effect is an effect size of zero. 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 confidence level is referred to as 95% or 99% guarantee of the intervals respectively.When there is a vast population data is available, confidence interval allows to choose correct interval for the research. 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.
It represents the range of numbers that are supposed to be good estimates for the population parameter.
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. It is very important to find confidence interval in order to perform any research or statistical survey on a large population data.

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