1. Understanding Basic Statistics
Chapter 8
Estimation
Understanding Basic Statistics
Fifth Edition
By Brase and Brase
Prepared by Jon Booze

2. Estimating µ When σ is Known

3. Point Estimate
An estimate of a population parameter given by a single number.

4. Margin of Error
Even if we take a very large sample size, may differ from µ.

5. Confidence Levels
A confidence level, c, is any value between 0 and 1 that corresponds to the area under the standard normal curve between –zc and +zc.

6. Critical Values

7. Critical Values
Which of the following correctly expresses the confidence interval shown at right? z
–2.58
2.58
a) b).
c) d).

8. Critical Values
Which of the following correctly expresses the confidence interval shown at right? z
–2.58
2.58
a) b).
c) d).

9. Common Confidence Levels

10. Recall From Sampling Distributions
If we take samples of size n from our population, then the distribution of the sample mean has the following characteristics:

12. Maximal Margin of Error
Since µ is unknown, the margin of error | – µ| is unknown.
Using confidence level c, we can say that differs from µ by at most:

13. The Probability Statement
In words, c is the probability that the sample mean will differ from the population mean by at most

14. Confidence Intervals

16. For a population of domesticated geese, the standard deviation of the mass is 1.3 kg. A sample of 45 geese has a mean mass of 5.7 kg.
Find the confidence interval for the population mean at the 95% confidence level.
a) <  < b). 0 <  < 2.97
c) <  < d) <  < 6.02

17. For a population of domesticated geese, the standard deviation of the mass is 1.3 kg. A sample of 45 geese has a mean mass of 5.7 kg.
Find the confidence interval for the population mean at the 95% confidence level.
a) <  < b). 0 <  < 2.97
c) <  < d) <  < 6.02

18. Critical Thinking Since is a random variable, so are the endpoints
After the confidence interval is numerically fixed for a specific sample, it either does or does not contain µ.

19. Critical Thinking
If we repeated the confidence interval process by taking multiple random samples of equal size, some intervals would capture µ and some would not!
The equation
states that the proportion of all intervals containing µ will be c.

20. Interpretation of the Confidence Interval
At the 0.90 confidence level, 1 in 10 samples, on average, will fail to enclose the true mean  within the confidence interval.

21. Estimating µ When σ is Unknown
In most cases, researchers will have to estimate σ with s (the standard deviation of the sample).
The sampling distribution for will follow a non-normal distribution called the Student’s t distribution.

22. The t Distribution

23. The t Distribution Find the t-value for the following data:
a). – b). –0.11
c). – d). –4.37

24. The t Distribution Find the t-value for the following data:
a). – b). –0.11
c). – d). –4.37

25. The t Distribution

26. The t Distribution
Use Table 6 of Appendix II to find the critical values tc for a confidence level c.
The figure to the right is a comparison of two t distributions and the standard normal distribution.

27. Using Table 6 to Find Critical Values
Degrees of freedom, df, are the row headings.
Confidence levels, c, are the column headings.

28. Using Table 4 to Find Critical Values
Use Table 4 in the Appendix to find the critical value tc for a 0.95 confidence level for a t-distribution with sample size n = 32.
a) b)
c) d)

29. Using Table 4 to Find Critical Values
Use Table 4 in the Appendix to find the critical value tc for a 0.95 confidence level for a t-distribution with sample size n = 32.
a) b)
c) d)

30. Maximal Margin of Error
If we are using the t distribution:

32. What Distribution Should We Use?

33. Estimating p in the Binomial Distribution
We will use large-sample methods in which the sample size, n, is fixed.
We assume the normal curve is a good approximation to the binomial distribution if both np > 5 and nq = n(1 – p) > 5.

34. Point Estimates in the Binomial Case

35. Margin of Error
The magnitude of the difference between the actual value of p and its estimate is the margin of error.

36. The Distribution of
For large samples, the distribution is well approximated by a normal distribution.

37. A Probability Statement
With confidence level c, as before.

39. Public Opinion Polls

40. Choosing Sample Sizes
When designing statistical studies, it is good practice to decide in advance:
The confidence level
The maximal margin of error
Then, we can calculate the required minimum sample size to meet these goals.

41. Sample Size for Estimating μ
If σ is unknown, use σ from a previous study or conduct a pilot study to obtain s.
Always round n up to the next integer!!

42. Sample Size for Estimating
If we have no preliminary estimate for p, use the following modification:

43. Sample Size for Estimating
How many students should be surveyed to determine the proportion of students who prefer vanilla ice cream to chocolate, accurate to 0.1 at a 90% confidence level?
a). 100 b). 69 c). 52 d). 5

44. Sample Size for Estimating
How many students should be surveyed to determine the proportion of students who prefer vanilla ice cream to chocolate, accurate to 0.1 at a 90% confidence level?
a). 100 b). 69 c). 52 d). 5