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

2. 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 )

3. 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

4. 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.

5. 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 s 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.

6. 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)

7. • 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)

8. 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

9. 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.

10. 2-Standard Error for the mean of paired differences

11. 3-Standard Error of the Difference between Two Sample Means

12. 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)

13. Finding The t-multiplier by using table a.2

14. 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.

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

16. 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* .

17. 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.

18. Example 11.9 Screen Time: Computer vs TV

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

20. 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).

21. 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= , t* = 2.18 (see Table A.2)

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

23. Example 11.14 Pooled t-Interval for Difference Between Mean Female and Male Sleep Times
1- df = n1+n2 – 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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