Full details can be found in the More Information Page on Transformations.
Z confidence interval for a mean
Another approach is bootstrapping. The simple formulae we have given above for the confidence interval of a mean only apply if you have used simple random sampling, or if individuals have been assigned to treatments at random. We consider what to do about clustered sampling and stratification in Unit 7.
You have only estimated your population mean using your sample mean - you have not measured it directly. Therefore, your confidence interval applies to the sample mean, not the population mean. Ideally your data should be drawn from a normally distributed population.
However, sample means of large numbers of observations tend to be distributed normally, whatever the underlying distribution. Hence the confidence interval may still be valid. But for very skewed distributions whatever the sample size, and for small samples from non-normal populations, you should first carry out an appropriate transformation of the data.
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- Confidence Interval.
- Confidence Intervals with the z and t-distributions.
You have only estimated the variability of your mean - you have not measured it directly. The values of t to be used in a confidence interval can be looked up in a table of the t distribution. A small version of such a table is shown in Table 1. The first column, df, stands for degrees of freedom, and for confidence intervals on the mean, df is equal to N - 1, where N is the sample size.
You can also use the " inverse t distribution " calculator to find the t values to use in confidence intervals. You will learn more about the t distribution in the next section. Assume that the following five numbers are sampled from a normal distribution: 2, 3, 5, 6, and 9 and that the standard deviation is not known. If we knew the population variance, we could use the following formula:. The next step is to find the value of t. We will finish with an analysis of the Stroop Data.
Specifically, we will compute a confidence interval on the mean difference score. Recall that 47 subjects named the color of ink that words were written in. The names conflicted so that, for example, they would name the ink color of the word " blue " written in red ink. The correct response is to say "red" and ignore the fact that the word is "blue. Table 2 shows the time difference between the interference and color-naming conditions for 10 of the 47 subjects. The mean time difference for all 47 subjects is Continuous Variable.
Dichotomous Variable. For both continuous variables e. Recall that sample means and sample proportions are unbiased estimates of the corresponding population parameters. For both continuous and dichotomous variables, the confidence interval estimate CI is a range of likely values for the population parameter based on:.
In practice, however, we select one random sample and generate one confidence interval, which may or may not contain the true mean. The confidence interval does not reflect the variability in the unknown parameter. 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. The Central Limit Theorem introduced in the module on Probability stated that, for large samples, the distribution of the sample means is approximately normally distributed with a mean:.
The Central Limit Theorem states that for large samples:.
Confidence Intervals with the z and t-distributions | Jacob Montgomery
Thus, the margin of error is 1. So, the general form of a confidence interval is:. Desired Confidence Interval. The t distribution is similar to the standard normal distribution but takes a slightly different shape depending on the sample size. In a sense, one could think of the t distribution as a family of distributions for smaller samples.
Instead of "Z" values, there are "t" values for confidence intervals which are larger for smaller samples, producing larger margins of error, because small samples are less precise. Just as with large samples, the t distribution assumes that the outcome of interest is approximately normally distributed.
A table of t values is shown in the frame below. Note that the table can also be accessed from the "Other Resources" on the right side of the page. 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. We select a sample and compute descriptive statistics including the sample size n , the sample mean, and the sample standard deviation s.
The formulas for confidence intervals for the population mean depend on the sample size and are given below. A point estimate for the true mean systolic blood pressure in the population is The margin of error is very small here because of the large sample size. Because the sample size is small, we must now use the confidence interval formula that involves t rather than Z.
Note that the margin of error is larger here primarily due to the small sample size. Suppose we wish to estimate the proportion of people with diabetes in a population or the proportion of people with hypertension or obesity. These diagnoses are defined by specific levels of laboratory tests and measurements of blood pressure and body mass index, respectively. Subjects are defined as having these diagnoses or not, based on the definitions. 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.
Recall that for dichotomous outcomes the investigator defines one of the outcomes a "success" and the other a failure. The sample size is denoted by n, and we let x denote the number of "successes" in the sample.
For example, if we wish to estimate the proportion of people with diabetes in a population, we consider a diagnosis of diabetes as a "success" i. If there are more than 5 successes and more than 5 failures, then the confidence interval can be computed with this formula:. The point estimate for the population proportion is the sample proportion, and the margin of error is the product of the Z value for the desired confidence level e. In other words, the standard error of the point estimate is:.
This formula is appropriate for large samples, defined as at least 5 successes and at least 5 failures in the sample. This was a condition for the Central Limit Theorem for binomial outcomes. If there are fewer than 5 successes or failures then alternative procedures, called exact methods, must be used to estimate the population proportion. The sample proportion is:.
This is the point estimate, i. The sample is large, so the confidence interval can be computed using the formula:. Specific applications of estimation for a single population with a dichotomous outcome involve estimating prevalence, cumulative incidence, and incidence rates.
- Sampling Distribution of the Sample Mean;
- Confidence Intervals!
- Confidence Intervals for Unknown Mean and Unknown Standard Deviation.
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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. For example, we might be interested in comparing mean systolic blood pressure in men and women, or perhaps compare body mass index BMI in smokers and non-smokers.
Both of these situations involve comparisons between two independent groups, meaning that there are different people in the groups being compared. We could begin by computing the sample sizes n 1 and n 2 , means and , and standard deviations s 1 and s 2 in each sample. The point estimate for the difference in population means is the difference in sample means:.
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. The standard error of the point estimate will incorporate the variability in the outcome of interest in each of the comparison groups. If we assume equal variances between groups, we can pool the information on variability sample variances to generate an estimate of the population variability.
Therefore, the standard error SE of the difference in sample means is the pooled estimate of the common standard deviation Sp assuming that the variances in the populations are similar computed as the weighted average of the standard deviations in the samples, i. If the sample sizes are larger, that is both n 1 and n 2 are greater than 30, then one uses the z-table.
For both large and small samples 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. These formulas assume equal variability in the two populations i. 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. If not, then alternative formulas must be used to account for the heterogeneity in variances. Next, we will check the assumption of equality of population variances.
The ratio of the sample variances is Notice that for this example Sp, the pooled estimate of the common standard deviation, is 19, and this falls in between the standard deviations in the comparison groups i. Therefore, the confidence interval is 0.
Our best estimate of the difference, the point estimate, is 1. The standard error of the difference is 0. Note that when we generate estimates for a population parameter in a single sample e. In contrast, when comparing two independent samples in this fashion the confidence interval provides a range of values for the difference.
In this example, we estimate that the difference in mean systolic blood pressures is between 0. In this example, we arbitrarily designated the men as group 1 and women as group 2. Had we designated the groups the other way i. The table below summarizes differences between men and women with respect to the characteristics listed in the first column. The second and third columns show the means and standard deviations for men and women respectively. Men have lower mean total cholesterol levels than women; anywhere from The men have higher mean values on each of the other characteristics considered indicated by the positive confidence intervals.
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. It is often of interest to make a judgment as to whether there is a statistically meaningful difference between comparison groups. This judgment is based on whether the observed difference is beyond what one would expect by chance. If there is no difference between the population means, then the difference will be zero i.
Zero is the null value of the parameter in this case the difference in means. If the confidence interval does not include the null value, then we conclude that there is a statistically significant difference between the groups. 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. Note, however, that some of the means are not very different between men and women e.
This means that there is a small, but statistically meaningful difference in the means. 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. The following table contains descriptive statistics on the same continuous characteristics in the subsample stratified by sex.
We will again arbitrarily designate men group 1 and women group 2. Since the sample sizes are small i. However,we will first check whether the assumption of equality of population variances is reasonable. The ratio of the sample variances is 9. The solution is shown below. Note that again the pooled estimate of the common standard deviation, Sp, falls in between the standard deviations in the comparison groups i. Interpretation: Our best estimate of the difference, the point estimate, is The standard error of the difference is 6. In this sample, the men have lower mean systolic blood pressures than women by 9.
Again, the confidence interval is a range of likely values for the difference in means. Since the interval contains zero no difference , we do not have sufficient evidence to conclude that there is a difference. The previous section dealt with confidence intervals for the difference in means between two independent groups.