| What Is a Confidence Interval? | | | | to get to the true population mean? |
| - We know that when we take the average of a | | | | - How well do you think this one item represents the |
| sample, it is probably not exactly the same as the | | | | true mean? |
| average of the population. | | | | - How much ability do we have to draw conclusions |
| - Confidence intervals help us determine the likely | | | | about the mean? |
| range of the population parameter. | | | | - What if we sample 900 items? Now, how close |
| * For example, if my 95% confidence interval is 5 +/- | | | | would we expect to get to the true population mean? |
| 2, then I have 95% confidence that the mean of the | | | | Three concepts affect the conclusions drawn from a |
| population is between 3 and 7. | | | | single sample data set of (n) items: |
| Why Do We Need Confidence Intervals? | | | | - Variation in the underlying population (sigma) |
| - Sample statistics, such as Mean and Standard | | | | - Risk of drawing the wrong conclusions (alpha, beta) |
| Deviation, are only estimates of the population's | | | | - How small a Difference is significant (delta) |
| parameters. | | | | - These 3 factors work together. Each affects the |
| - Because there is variability in these estimates from | | | | others. |
| sample to sample, we can quantify our uncertainty | | | | Variation: When there's greater variation, a larger |
| using statistically-based confidence intervals. | | | | sample is needed to have the same level of |
| - Confidence intervals provide a range of plausible | | | | confidence that the test will be valid. More variation |
| values for the population parameters (m and s). | | | | diminishes our confidence level. |
| - Any sample statistic will vary from one sample to | | | | Risk: If we want to be more confident that we are not |
| another and, therefore, from the true population or | | | | going to make a decision error or miss a significant |
| process parameter value. | | | | event, we must increase the sample size. |
| Confidence Interval for the Mean (Mu) with Population | | | | Difference: If we want to be confident that we can |
| Standard Deviation (Sigma) Unknown: | | | | identify a smaller difference between two test |
| - A very important point to remember is that for this | | | | samples, the sample size must increase. |
| example we assumed that we knew the population | | | | - Larger samples improve our confidence level. |
| standard deviation, and many times that is not the | | | | - Lower confidence levels allow smaller samples. |
| case. Often, we have to estimate both the mean and | | | | - All of these translate into a specific confidence |
| the standard deviation from the sample. | | | | interval for a given parameter, set of data, confidence |
| - When Sigma is not known, we use the t-distribution | | | | level and sample size. |
| rather than the normal (z) distribution. | | | | - They also translate into what types of conclusions |
| The t-distribution: | | | | result from hypothesis tests. |
| - In many cases, the true population Sigma is not | | | | - Testing for larger differences between the samples, |
| known, so we must use our sample standard deviation | | | | reduces the size of the sample. This is known as delta |
| (s) as an estimate for the population standard deviation | | | | (D). |
| (s). | | | | Type I Error |
| - Since there is less certainty (not knowing Mu or | | | | - Alpha Risk or Producer Risk is the risk of rejecting |
| Sigma ), the t-distribution essentially "relaxes" or | | | | the null, and taking action, when none was necessary |
| "expands" our confidence intervals to allow for this | | | | - It is the alpha value you choose. |
| additional uncertainty. | | | | - The confidence level is one minus the alpha level. |
| - In other words, for a 95% confidence interval, you | | | | - Most non-critical business processes choose an |
| would multiply the standard error by a number greater | | | | alpha of 5% with a Confidence Level of 95 |
| than 1.96, depending on the sample size. | | | | Type II Error |
| - 1.96 comes from the normal distribution, but the | | | | - Beta Risk or - Consumer Risk is the risk of accepting |
| number we will use in this case will come from the | | | | the null when you should have rejected it. |
| t-distribution. | | | | No action is taken when there should have been |
| What Is This t-Distribution? | | | | action. |
| - The t-distribution is actually a family of distributions. | | | | - The Type II Error is determined from the |
| - They are similar in shape to the normal distribution | | | | circumstances of the situation. |
| (symmetric and bell-shaped), although wider, and flatter | | | | - If alpha is made very small, then beta increases (all |
| in the tails. | | | | else being equal). |
| - How wide and flat the specific t-distribution is | | | | - Requiring overwhelming evidence to reject the null |
| depends on the sample size. The smaller the sample | | | | increases the chances of a type II error. |
| size, the wider and flatter the distribution tails. | | | | - To minimize beta, while holding alpha constant, |
| - As sample size increases, the t-distribution | | | | requires increased sample sizes. |
| approaches the exact shape of the normal distribution. | | | | - One minus beta is the probability of rejecting the null |
| Sample Size Concerns | | | | hypothesis when it is false. This is referred to as the |
| - If we sample only one item, how close do we expect | | | | Power of the test. |