Estimating Means With Confidence Prepared by Wrya H.Nadir Do men lose more weight from dieting or from exercising?

content 1- introduction 2-Recall from Chapter 10 3-Estimation Situations Involving Means module 3 ,4,5 4-Standard Errors 5-Student’s t-Distribution: 6-Degrees of Freedom 7- Confidence Interval for the Difference in Two Population Means (Independent Samples) ( 2-pooled )

introduction -In Chapter 10, we learned how to form and interpret a confidence interval for a population proportion and for the difference in two population proportions. Remember that a confidence interval is an interval of values computed from sample data that is likely to include the true population value. -Here we will review material from Chapters 9 and 10 that is relevant to finding confidence intervals for parameters involving means 1- parameter 2-A statistic, or estimate, is a characteristic of a sample. A statistic estimates a parameter

Recall from Chapter 10 - A confidence interval or interval estimate for any of the five parameters can be expressed as where the multiplier is a number based on the confidence level desired and determined from the standard normal distribution (for proportions) or Student’s t-distribution (for means). - We will discuss each of the three components of this formula for the three situations involving means.

Estimation Situations Involving Means Situation 1. Estimating the mean of a quantitative variable • Example research questions: 1- What is the mean amount of time that college students spend watching TV per day? 2-What is the mean number of emails students get per day? -Population parameter: μ (spelled “mu” and pronounced “mew”) = population mean for the variable -Sample estimate: x = the sample mean for the variable, based on a sample of size n.

Example research questions: Situation 2: Estimating the Population Mean of Paired Differences for a Quantitative Variable Example research questions: 1-What is the mean difference in age between husbands and wives in Britain? 2-What is the mean difference in blood pressure before and after meditation? 3- What is the mean difference in hrs/day studying and online (for non-study reasons) for college students? -Dependent/paired sample: scores are linked (same person ,married couple)

• Example research questions: Situation 3: Estimating the Difference between Two Population Means for a Quantitative Variable (Independent Samples) • Example research questions: 1-How much difference is there between the mean foot lengths of men and women? 2- Test score from Prof. X class , Test score from Prof .Y class 3-Males heights, Females heights -Independent sample: score are separated (different people)

Standard Errors -The standard error of a sample statistic measures, roughly, the average difference between the statistic and the population parameter. This “average difference” is over all possible random samples of a given size that can be taken from the population. -The standard error gives a measure of how well a sample represents the population. Different sample in same population give different result

1-Standard Error of a Sample Mean Example 11.2 Mean Hours per Day That Penn State Students Watch TV The sample mean=2.09hours per day, is the statistic that estimates mu, the population mean.

2-Standard Error for the mean of paired differences

3-Standard Error of the Difference between Two Sample Means

Student’s t-Distribution: If the sample size n is small, this standardized statistic will not have a N(0,1) distribution but rather a t-distribution with n – 1 degrees of freedom (df). Page(342)(ch9)

Finding The t-multiplier by using table a.2

CI Module 3: Confidence Intervals for One Population Mean -Condition: 1-Population of measurements is bell-shaped (no major skewness or outliers) 2- Population of measurements is not bell-shaped, but a large random sample is measured, n ≥ 30.

Ex11.5(p414) Interpretation: We can say with 95% confidence that in the population represented by the sample, the mean forearm length is between 24.33 and 26.67 cm.

Special Case: Approximate 95% Confidence Intervals for Large Samples 1- for large values of degrees of freedom, the t-distribution is very close to the standard normal distribution. 2-t* multiplier will be close to the corresponding z* multiplier 3-For 95% confidence, the z* multiplier is 1.96 but is often rounded off to 2. 4-The t* multiplier for df=1000 is 1.96 as well; for df=100, it is 1.98; and for df= 70, 80, or 90, t* is 1.99, which is still close to the exact z* value of 1.96. 5-Even with as few as 30 df, the t* multiplier is 2.04, not far from the exact z* of 1.96 and even closer to the rounded-off value of 2.0. Therefore, when we have a large sample, we can simplify matters and find an approximate 95% confidence interval estimate using a multiplier of 2.0 instead of t* .

CI Module 4: Confidence Interval for the Population Mean of Paired Differences 1-Data: two variables for each of n individuals or pairs; use the difference d= x1 - x2 2- Population parameter: (md) = mean of differences for the population, equivalent to (m1 - m2 ) 3-Sample estimate: d= sample mean of the differences, equivalent to (x1-x2). 4- 5-Condition: 1-Population of measurements is bell-shaped (no major skewness or outliers) 2- Population of measurements is not bell-shaped, but a large random sample is measured, n ≥ 30.

Example 11.9 Screen Time: Computer vs TV

CI Module 5: 1-Confidence Interval for the Difference in Two Population Means (Independent Samples) ( 1-unpooled)

Degrees of Freedom The t-distribution is only approximately correct and df formula is complicated (Welch’s approx): Statistical software can use the above approximation, but if done by hand then use a conservative df = smaller of (n1 – 1) and (n2 – 1).

Example 11.11 Effect of a Stare on Driving The 95% confidence interval for the difference between the population means is 0.14 seconds to 1.93 seconds If software not available the minimum of the two values (n1-1) and (n2 - 1), which is df= 13 –1=12. , t* = 2.18 (see Table A.2)

Confidence Interval for the Difference in Two Population Means (Independent Samples) ( 2-pooled)

Example 11.14 Pooled t-Interval for Difference Between Mean Female and Male Sleep Times 1- df = n1+n2 – 2 =83+65-2=146 2- If Table A.2 is used, estimate of the multiplier t* is 1.98 (for df = 100 and 0.95 confidence) 3- 4-

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