12 - Confidence Intervals

Comparing multiple groups with confidence intervals

- If two confidence intervals do not overlap, then those two groups are significantly different - If the two intervals do overlap, then those two groups are not significantly different - Confidence intervals can also be displayed visually

Lower and Upper limit of a two-sided confidence interval

- Lower: The sample mean minus the margin - Upper: The sample mean plus the margin Interval is written in the format (lower limit, upper limit) CI = x̄ ± m

Note: that based on our margin of error formula, for a two-sided test ...

-A larger standard deviation will lead to a larger margin and length -A larger sample size will lead to a smaller margin and length - The larger your confidence level, the larger your margin and length, since a larger confidence results in a large critical test statistic

Interval estimates

Are a range of possible values for that estimate

Point estimates

Are the calculation of a single statistic, such as a sample mean x̄

Estimation

Is another closely related method, generally used in tandem with hypothesis testing, to make statistical inferences about a population. - This involves generating point and interval estimates

95% CI

If we were to repeat our sampling method 100 times, 05 intervals would cover the true population mean - note this is NOT a probability

Line Placement study Example 1

In the previous study of recovery time among PICC line patients, our sample of 60 had a mean recovery time of 4.1 days. The population of Port-a-Cath patients we tested against had a mean μ=4.7 and standard deviation σ=2.2. We used a two-sided, one-sample Z test to test if our sample had a different recovery time from 4.7. We can also use estimation for this purpose.

Confidence

Represents your study's ability to correctly retain the null when the null is true. It is based on alpha, so it is also predetermined Confidence =1-α =P(retain H_0 | H_0 is true)

Line Placement Length Example

Since we have already calculated the margin of error as 0.557, we can easily calculate the length of the interval: L =2*0.557 =1.134

Length of a two-sided interval

This is 2 times the margin. It is also the interval upper limit minus its lower limit.

Margin of error

This is the distance from the center (point estimate) of the interval to it's ends - A function of the standard error of the mean and the critical test statistic, based on our alpha - When using a population σ, we can calculate the margin of error for a two-sided test from the standard error of the mean and the critical Z: m=(Z_1-α/2)*(σ/√n)

Confidence intercal (CI) of the mean

Used to make an interval estimation of the population mean μ from sample data. An α of 0.05 results in a 95% confidence interval. An α of 0.01 results in a 99% confidence interval.

Line Placement Study Comparison Example

We are 95% confident the interval (3.543, 4.657) covers the true population mean recovery time for PICC line patients. Lets say a sample of tunneled central line patients yielded a 95% confidence interval of (4.922, 6.465). Since these two intervals do not overlap, the recovery time of the two groups are significantly different.

Line Placement Study CI Example

We can estimate a 95% confidence interval for the true mean recovery time for PICC line patients

Line Placement Margin of error example

We can estimate the margin of error for a 95% confidence interval of the true population mean. For a two-sided Z test at alpha=0.05, the critical Z is 1.96. The σ=2.2 and the sample size is 60. m =1.96*(2.2/√60) =1.96*(0.384) =0.557

To calculate a confidence interval

We must first calculate margin of error

Making inferences

hypothesis testing is the standard method of making inferences from a sample to a population

Line Placement Upper Lower CI Example

since our sample mean was 4.1 and our margin of error was 0.557, our confidence interval is 4.1 ± 0.557/ 4.1-0.557=3.543 , 4.1+.557=4.657 95% CI = (3.543,4.657) We are 95% confident the interval (3.543,4.657) covers the true population mean recovery time for PICC line patients. Since this does not overlap the hypothesized population mean μ=4.7, we can reject the null hypothesis. The mean recovery time of PICC line patients is significantly lower than that of Port-a-Cath patients