Confidence intervals explanation and example
Chapter 6 Introduction to Inferential Statistics
Confidence Levels
Our First Statistical Test We are now ready to put what we have learned about sampling, sampling distributions, and the standard error of the mean to begin inferential statistics. If a sample is drawn from a population, the sample mean value is the best estimate of the population mean for that variable. (point estimate) However, we realize that not every sample mean 𝑋 will be exactly the same as the population mean (μ).
How to Make a Confidence Interval Using what we know about the standard error of the mean and the normal distribution we can construct an interval around our point estimate and even attach to it a level of confidence (CI) that the interval contains the population mean. Use this equation to construct a confidence interval: 𝑋 − (𝑧 ∗ 𝜎𝑋 ) to 𝑋 + (𝑧 ∗ 𝜎𝑋 )
How to Make a Confidence Interval (Cont.) In words, we add to and subtract from the sample mean, a value that is determined by multiplying the standard error of the mean by a z-value. The z-value selected determines the amount of confidence we have that the population mean lies within the range of our interval. Z for 95% CI = 1.96 Z for 99% CI = 2.58
𝑋 − 𝑧 ∗ 𝜎𝑋 to 𝑋 + 𝑧 ∗ 𝜎𝑋
Example The measured resting heart rate for a sample of 16, twelve year old boys, is 77. Our best estimate of the mean resting heart rate for the population of boys from which this sample was drawn is 77. From published data we know that the mean and standard deviation of resting heart rates for twelve year old boys are 𝜇 = 80 𝑎𝑛𝑑 𝜎 = 6.5.
Example (Cont.) The standard error of the mean is
6.5 16
= 1.625
The 95% confidence interval is calculated to be: 77-(1.96*1.625) to 77+(1.96*1.625) = 77 ± 3.185 = 73.815 to 80.185 We can now use this confidence interval to come to the conclusion that the mean resting heart rate of our sample (77), is not statistically different than the national mean because the national mean of 80 lies inside the interval.
Example (Cont.)
77 73.815
95% CI
80 80.185
Confidence Levels
Our First Statistical Test We are now ready to put what we have learned about sampling, sampling distributions, and the standard error of the mean to begin inferential statistics. If a sample is drawn from a population, the sample mean value is the best estimate of the population mean for that variable. (point estimate) However, we realize that not every sample mean 𝑋 will be exactly the same as the population mean (μ).
How to Make a Confidence Interval Using what we know about the standard error of the mean and the normal distribution we can construct an interval around our point estimate and even attach to it a level of confidence (CI) that the interval contains the population mean. Use this equation to construct a confidence interval: 𝑋 − (𝑧 ∗ 𝜎𝑋 ) to 𝑋 + (𝑧 ∗ 𝜎𝑋 )
How to Make a Confidence Interval (Cont.) In words, we add to and subtract from the sample mean, a value that is determined by multiplying the standard error of the mean by a z-value. The z-value selected determines the amount of confidence we have that the population mean lies within the range of our interval. Z for 95% CI = 1.96 Z for 99% CI = 2.58
𝑋 − 𝑧 ∗ 𝜎𝑋 to 𝑋 + 𝑧 ∗ 𝜎𝑋
Example The measured resting heart rate for a sample of 16, twelve year old boys, is 77. Our best estimate of the mean resting heart rate for the population of boys from which this sample was drawn is 77. From published data we know that the mean and standard deviation of resting heart rates for twelve year old boys are 𝜇 = 80 𝑎𝑛𝑑 𝜎 = 6.5.
Example (Cont.) The standard error of the mean is
6.5 16
= 1.625
The 95% confidence interval is calculated to be: 77-(1.96*1.625) to 77+(1.96*1.625) = 77 ± 3.185 = 73.815 to 80.185 We can now use this confidence interval to come to the conclusion that the mean resting heart rate of our sample (77), is not statistically different than the national mean because the national mean of 80 lies inside the interval.
Example (Cont.)
77 73.815
95% CI
80 80.185
