Download presentation

Presentation is loading. Please wait.

Published byEsperanza Mudd Modified over 4 years ago

1
Confidence Intervals for a Mean when you have a “large” sample…

2
The situation Want to estimate the actual population mean . But can only get , the sample mean. Find a range of values, L < < U, that we can be really confident contains . This range of values is called a “confidence interval.”

3
Confidence Intervals for Proportions in Newspapers 18% of women, aged 18-24, think they are overweight. The “margin of error” is 5%. The “confidence interval” is 18% ± 5%. We can be really confident that between 13% and 23% of women, aged 18-24, think they are overweight.

4
General Form of most Confidence Intervals Sample estimate ± margin of error Lower limit L = estimate - margin of error Upper limit U = estimate + margin of error Then, we’re confident that the population value is somewhere between L and U.

5
Example Let X = number of high school friends Stat 250 students keep in touch with. True population mean = 5 friends. True population standard deviation = 5 friends. Take a random sample of 36 Stat 250 students. Calculate .

6
Sampling Distribution of 55 + 2(0.83) 6.7 5 - 2(0.83) 3.3 0.95

7
What does the sampling distribution tell us? 95% of the sample means will fall within 2 standard errors, or within 1.66 friends, of the true population mean = 5. Or, 95% of the time, the true population mean = 5 will fall within 2 standard errors, or within 1.66 friends, of the sample mean.

8
Sampling Distribution of + 2( / n) - 2( / n) 0.95

9
What does the sampling distribution tell us? 95% of the sample means will fall within 2 standard errors of the population mean Or, 95% of the time, the true population mean will fall within 2 standard errors of the sample mean. Use this last statement to create a formula.

10
95% Confidence Interval for Formula in notation: Formula in English: Sample mean ± (2 × standard error of the mean)

11
95% Confidence Interval for Formula in notation: Formula in English: Sample mean ± (2 × estimated standard error) 1. Formula OK as long as sample size is large (n 30) 2. Margin of error = 2 × standard error of the mean 3. 95% is called the “confidence level”

12
Example A random sample of 32 students reported combing their hair an average of 1.6 times a day with a standard deviation of 1.3 times a day. In what range of values can we be 95% confident that , the actual mean, falls?

13
What does 95% confident mean? + 2( / n) - 2( / n) 0.95 95% of all such confidence intervals will contain the true mean

14
What if you want to be more (or less) confident? + Z( / n) - Z( / n) 0.98 1. Put confidence level in middle. 2. Subtract from 1. 3. Divide by 2 and put in tails. 4. Look up Z value. 0.01 Z 0.99 = 2.33

15
Any % Confidence Interval for Formula in notation: Formula in English: Sample mean ± (Z × estimated standard error)

16
Example A random sample of 64 students reported having an average of 2.4 roommates with a standard deviation of 4 roommates. In what range of values can we be 96% confident that , the actual mean, falls?

17
Length of Confidence Interval Want confidence interval to be as narrow as possible. Length = Upper Limit - Lower Limit

18
How length of CI is affected? As sample mean increases… As the standard deviation decreases… As we decrease the confidence level… As we increase sample size …

19
Warning #1 Confidence intervals are only appropriate for random, representative samples. Problematic samples: –magazine surveys –dial-in surveys (1-900-vote-yes) –internet surveys (CNN QuickVote)

20
Warning #2 The confidence interval formula we learned today is only appropriate for large samples (n 30). If you use today’s formula on a small sample, you’ll get a narrower interval than you should. Will learn correct formula for small samples.

21
Warning #3 The confidence interval for the mean is a range of possible values for the population average. It says nothing about the range of individual measurements. The empirical rule tells us this.

Similar presentations

OK

Introduction Studying the normal curve in previous lessons has revealed that normal data sets hover around the average, and that most data fits within.

Introduction Studying the normal curve in previous lessons has revealed that normal data sets hover around the average, and that most data fits within.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google