1. Chapter 8: Interval estimation of parameters
Mr. Mark Anthony Garcia, M.S.
Mathematics Department
De La Salle University
Chapter 8: Interval estimation of parameters

2. Recall: Parameter
A parameter is a numerical value associated to a population.
Some examples are the population mean 𝜇, population standard deviation 𝜎, population variance 𝜎 2 and population proportion 𝑝.

3. Recall: Statistic
A statistic is a numerical value associated to a sample.
Some examples are the sample mean 𝑥 , sample standard deviation 𝑠, sample variance 𝑠 2 and sample proportion 𝑝 .

4. Estimation
In this chapter, we are concerned in estimating the value of the population parameter using the value of the sample statistic.
In particular, we estimate the value of the population proportion 𝑝 using the value of the sample proportion 𝑝 .

5. Estimation
Estimation is a procedure by which a numerical value or values are assigned to a population parameter based on the information collected from a sample.

6. Examples of using Estimation
A political organization may want to estimate the proportion of voters that will vote for their presidential candidate.
A TV network may want to estimate the proportion of TV viewers who watch a particular primetime show.

7. Situation: Estimation
Suppose that a political analyst wants to estimate the percentage or proportion of voters that will vote for senatorial candidate A in city X for the upcoming senatorial elections.
The population of study of the political analyst is the set of all voters in city X. Suppose that the population size is 𝑁=1000 voters.

8. Situation: Estimation
The political analyst used simple random sampling to obtain a sample of 𝑛=50 voters and asked these voters if they will vote for senatorial candidate A or not.
The survey result showed that 34 out of the 50 voters will vote for senatorial candidate A.

9. Situation: Estimation
The result showed that the sample proportion 𝑝 = =0.68 or 68% of the sample will vote for senatorial candidate A.
However, the result is obtained from a sample of 𝑛=50 voters. The sample that the political analyst obtained is only one possible sample of 50 voters out of many different sample of 50 voters.

10. Situation: Estimation
If the political analyst takes another sample of 50 voters (common voters from the previous sample may be included in the new sample), he will get a different sample proportion or percentage.

11. Situation: Estimation
The problem of the political analyst is to estimate the value of the population proportion 𝑝 using the different sample proportions obtained from different samples.
The question is, what percentage of the 𝑁=1000 voters in city X will vote for senatorial candidate A for the upcoming elections?

12. Situation: Estimation
To answer the problem of the political analyst, we use interval estimation of parameters.
But before we reveal this topic, let us differentiate point estimation from interval estimation.

13. Point Estimation
Point estimation is the procedure of assigning a single value to the population parameter.
Normally, we use the value of the sample statistic as point estimate for the population parameter.
Disadvantage: We can have a lot of point estimates since different samples may be obtained from the population.

14. Example: Point Estimation
In the situation of the political analyst, we can use the sample proportion 𝑝 = =0.68 or 68% as a point estimate for the population proportion 𝑝.

15. Interval Estimation
Interval estimation is the procedure of assigning multiple values to a population parameter.
It is a procedure of obtaining an interval where the population parameter coincides.

16. Interval Estimation
The interval is dependent on the point estimate, standard error and maximum error.
Each interval is constructed with regard to a given confidence level and is called a confidence interval.

17. Confidence Level
The confidence level attached to a confidence interval is viewed as the probability that the population parameter lies inside the confidence interval.
It is of the form 1−𝛼 100% .
If the confidence level is 95%, then it is of the form 1− % = % =95%. This means that 𝛼=0.05.

18. Confidence Level
The value of 𝛼 is viewed as the probability that the population parameter is not inside the confidence interval.
Now, let us interpret the 95% confidence level using the political analyst situation.

19. Confidence Level
Suppose that the political analyst obtained 100 samples of 50 voters each. Then 95 out of the 100 samples will give us a sample proportion that is inside the confidence interval estimate.

20. Interval Estimation of Proportion
The sample proportion is given by 𝑝 = 𝑥 𝑛 where 𝑥 is the number of successes in the sample and 𝑛 is the sample size.
The standard error is given by the formula 𝑆𝐸= 𝑝 𝑞 𝑛 where 𝑞 =1− 𝑝 . This can be obtained from the PhStat output.

21. Interval Estimation of Proportion
The formula for the confidence interval estimate for the population proportion 𝑝 is sample proportion ± maximum error.
We use minus for the lower bound and plus for the upper bound.

22. Interval Estimation of Proportion

23. Interval Estimation of Proportion
Thus, we are 95% confident that the proportion of voters that will vote for senatorial candidate A is between the interval [0.5507, ] or between 55.07% and 80.93%.

24. Example 1: Interval Estimation
In a random sample of 500 people eating lunch at a hospital cafeteria on various Fridays, it was found that 160 preferred seafood.
Find a 95% confidence interval for the actual proportion of people who prefer seafood on Fridays at this cafeteria.
How large a sample is required if we want to be 95% confident that our estimate is within 0.02?

25. Example 1: Interval Estimation
Find a 95% confidence interval for the actual proportion of people who prefer seafood on Fridays at this cafeteria.
Given: 𝑝 = =0.32, confidence level is 95%.

26. Example 1: Interval Estimation
Data
Sample Size
500
Number of Successes
160
Confidence Level
95%
Intermediate Calculations
Sample Proportion
0.32
Z Value
Standard Error of the Proportion
Interval Half Width
Confidence Interval
Interval Lower Limit
Interval Upper Limit
|𝒛 𝜶 𝟐 |=𝟏.𝟗𝟔
Maximum error

27. Example 1: Interval Estimation
We are 95% confident that the proportion of people eating lunch at a hospital cafeteria on various Fridays who prefer seafood is between [0.2791,0.3609] or between 27.91% and 36.09%

28. Sample Size Determination
If the sample proportion 𝑝 is used as an estimate for 𝑝, then we can be (1-α)100% confident that the error will not exceed a specified maximum error 𝑒 when the sample size is 𝑛= 𝑧 𝛼 2 𝑝 𝑞 𝑒 2

29. Example 1: Interval Estimation
How large a sample is required if we want to be 95% confident that our estimate is within 0.02?
The maximum error 𝑒=0.02 is given.
The problem is interpreted as the number of samples to be considered so that the error will not exceed 0.02.

30. Example 1: Interval Estimation
Data
Estimate of True Proportion
0.32
Sampling Error
0.02
Confidence Level
95%
Intermediate Calculations
Z Value
Calculated Sample Size
Result
Sample Size Needed
2090

31. Example 2: Interval Estimation
In a random sample of 1000 homes in a certain city it is found that 228 are heated by oil. Find a 99% confidence interval for the proportion of homes in this city that are heated by oil.

32. Example 2: Interval Estimation
Data
Sample Size
1000
Number of Successes
228
Confidence Level
99%
Intermediate Calculations
Sample Proportion
0.228
Z Value
Standard Error of the Proportion
Interval Half Width

33. Example 2: Interval Estimation
The lower and upper bounds are not given.
Thus, we use the formula sample proportion ± maximum error.
In the output, the maximum error is the interval half width.
Hence, we have sample proportion ± interval half width.

34. Example 2: Interval Estimation
Thus, we get 0.228±
Therefore, we are 99% confident that the proportion of all homes in that city that are heated by oil is between [0.1938,0.2622] or between 19.38% or 26.22%