Confidence Intervals for the One Sample Problem (Population Mean)

Transcription

1 Confidence Intervals for the One Sample Problem (Population Mean)

2 Outline 1 Introduction 2 Confidence Interval Basics 3 Case Study:Body Temperature

3 Outline 1 Introduction 2 Confidence Interval Basics 3 Case Study:Body Temperature

4 Readings

5 The Basic Problem Part of a study on the development of the thymus gland: weights of thymus gland from 5 chick embryos after 14 days of incubation: (mg) We want to know µ, the mean weight of thymus glands in the entire population of chick embryos after 14 days of incubation (in the same incubator). ȳ = is our best point estimate for µ. Goal: use the sampling distribution of Ȳ to determine an interval estimate.

6 Recall This is part two of the lecture We already studied (on the board) 1 The t distribution 2 How to compute a confidence interval for µ 3 Planning a study: sample size calculation

7 Lecture Goals 1 General: learn the vocabularly (ingredients) of confidence intervals 2 Specific: learn how to compute and interpret confidence intervals in the specific setting of the one sample population mean problem. 3 (later you will apply your general knowledge to understanding other specific settings).

8 Outline 1 Introduction 2 Confidence Interval Basics 3 Case Study:Body Temperature

9 Ingredients (General) 1 Initial estimate 2 Standard error 3 Sampling Distribution

10 Ingredients (Specific, µ) 1 Initial estimate: ȳ 2 Standard error: SEȳ = s n 3 Sampling Distribution: t(n 1)

11 Steps (General) Operationally, you should think 1 Compute the initial estimate. 2 Compute the standard error 3 Find the cutoff point from the sampling distribution, t 4 Combine the basic ingredients to form the confidence interval initial estimate ± t SE 5 Interpret and report the results.

12 Steps (Specific, µ) Operationally, you should think 1 Compute the initial estimate: ȳ 2 Compute the standard error: s n 3 Find cutoff points from the sampling distribution t(n 1): t 4 Combine the basic ingredients to form the confidence interval ȳ ± t 5 Interpret and report the results. s n

13 Return to the Thymus Gland Part of a study on the development of the thymus gland: weights of thymus gland from 5 chick embryos after 14 days of incubation: (mg) Summary: n = 5, ȳ = 31.72, s = 8.73 Do: Compute a 90% confidence for µ, the mean weight of thymus glands in the entire population of chick embryos after 14 days of incubation

14 Steps for this example 1 Compute the initial estimate: ȳ = Compute the standard error: s n = = Find cutoff points from the sampling distribution t(4): t = Combine the basic ingredients to form the confidence interval ± 2.132(3.90) = (23.41, 40.03) 5 Interpret and report the results.

15 Conclusion We are 90% confident that the mean thymus gland weight of chick embryos at the age of 14 days of incubation (in the condition of the experiment) is between and mg.

16 Outline 1 Introduction 2 Confidence Interval Basics 3 Case Study:Body Temperature

17 Case Study Example Body temperature varies within individuals over time (it can be higher when one is ill with a fever, or during or after physical exertion). However, if we measure the body temperature of a single healthy person when at rest, these measurements vary little from day to day, and we can associate with each person an individual resting body temperture. There is, however, variation among individuals of resting body temperture. A sample of n = 130 individuals had an average resting body temperature of degrees Fahrenheit and a standard deviation of 0.68 degrees Fahrenheit.

18 Case Study: Questions Example How can we use the sample data to estimate with confidence the mean resting body temperture in a population? Specifically,based on this data find 95% and 99% confidence intervals for the mean body temperature.

19 Calculations Example The sample mean and standard deviation from the n = 130 observations are ȳ = and s = There are 129 degrees of freedom; Table 4 does not have this value, so we round down to 100 The critical value for a 95% confidence interval is 1.984; The critical value for a 99% confidence interval is 2.626;

20 Calculations Continued Example The margin of error for the 95% confidence interval is / 130. = The margin of error for the 99% confidence interval is / 130. = The 95% confidence interval is < µ < The 99% confidence interval is < µ < We summarize the 99% confidence interval in context. We are 99% confident that the mean resting body temperature of healthy adults is between and degrees Fahrenheit. It is noteworthy that 98.6 is not in this interval.

21 Conclusion We summarize the 99% confidence interval in context of the study. We are 99% confident that the mean resting body temperature of healthy adults is between and degrees Fahrenheit.