3. How large a sample of one-bedroom apartments above would be needed to estimate the population mean within plus or minus $50 with 90% confidence? A confidence interval is a range of values, derived from sample statistics, that is likely to contain the value of an unknown population parameter. Most frequently, you’ll use confidence intervals to bound the mean or standard deviation, but you can also obtain them for regression coefficients, proportions, rates of occurrence (Poisson), and for the differences between populations.

As you increase the sample size, the sampling error decreases and the intervals become narrower.

Confidence intervals only tell you about the parameter of interest and nothing about the distribution of individual values. In the light bulb example, we know that the mean is likely to fall within the range, but the 95% confidence interval does not predict that 95% of future observations will fall within the range. A prediction interval is a type of confidence interval that you can use with predictions from linear and nonlinear models.

A confidence interval of the prediction is a range that is likely to contain the mean response given specified settings of the predictors in your model. Going back to our light bulb example, suppose we design an experiment to test how different production methods (Slow or Quick) and filament materials (A or B) affect the burn time.

A prediction interval is a range that is likely to contain the response value of a single new observation given specified settings of the predictors in your model.

The prediction interval is always wider than the corresponding confidence interval of the prediction because of the added uncertainty involved in predicting a single response versus the mean response. We’re getting down to determining where an individual observation is likely to fall, but you need a model for it to work.

A tolerance interval is a range that is likely to contain a specified proportion of the population.

In contrast, the width of a tolerance interval is due to both sampling error and variance in the population. Unfortunately, the percentile estimates will have error because we are working with a sample.

In general, use tolerance intervals if you have sampled data and want to predict a range of likely outcomes. With Minitab statistical software, it’s easy to obtain all of these intervals for your data!

Does this mean that confidence intervals give a range of MEANS while prediction intervals give a range of Y values? Prediction intervals give a range for the y-value of the next observation given specific x-values. A confidence interval is an estimate of a range of values that includes the true population value for a statistic, such as a mean.

In the example below, ten participants were asked to complete a task and the time in seconds was recorded.

In this example, alpha is set to 0.05 (a 95% confidence level), the standard deviation is calculated using the Excel STDEV function and the sample size is calculated using the Excel COUNT function.

Often times you may want to display your data in graphical form, including the confidence interval.

Which of the assumptions listed above might be problematic in making inference to the population in Question 4? Because of their random nature, it is unlikely that two samples from a given population will yield identical confidence intervals. If you could increase the sample size to equal the population, there would be no sampling error. There are two types of prediction intervals that use predictor values entered into the model equation. Just like the regular confidence intervals, the confidence interval of the prediction presents a range for the mean rather than the distribution of individual data points. After we fit a model, statistical software like Minitab can predict the response for specific settings.

We can be 95% confident that this range includes the mean burn time for light bulbs manufactured using these settings.

We can be 95% confident that this range includes the burn time of the next light bulb produced with these settings.

To generate tolerance intervals, you must specify both the proportion of the population and a confidence level. As the sample size approaches the entire population, the width of the confidence interval approaches zero.

As the sample size approaches the entire population, the sampling error diminishes and the estimated percentiles approach the true population percentiles.

We can’t be 100% confident that a tolerance interval truly contains the specified proportion.

In this context, tolerance intervals can detect excessive variation by comparing client requirements to tolerance limits that cover a specified proportion of the population.

The level of confidence you choose will determine how reasonably certain the true population mean is included in the range. Using the CONFIDENCE function in Excel, we can calculate the confidence interval of the sample data to estimate the range that contains the population mean. The chart below shows the mean of the data set above and graphs the confidence interval 13.02 as an error bar. But if you repeated your sample many times, a certain percentage of the resulting confidence intervals would contain the unknown population parameter.

The confidence interval indicates that you can be 95% confident that the mean for the entire population of light bulbs falls within this range.

In this case, the confidence interval would have a width of zero and be equal to the true population parameter.

We want to predict the mean burn time for bulbs that are produced with the Quick method and filament type A. However, it doesn’t tell us anything about the distribution of burn times for individual bulbs.

The manufacturer is 95% confident that at least 95% of all burn times will fall between 1060 to 1435 hours. If the tolerance interval is wider than the client's requirements, there may be too much product variation.

A confidence level of 95% means that you can be 95% certain, or wrong 5% of the time, that the sample being analyzed contains the population mean.

For example, a confidence level of 95% has an alpha of 5%, in other words, alpha is equal to 1 minus the confidence level.

In this post, we’ll take a look at the different types of intervals that are available in Minitab, their characteristics, and when you should use them.

The percentage of these confidence intervals that contain the parameter is the confidence level of the interval.

If this range is wider than their clients' requirements, the process may produce excessive defects.

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