INTERVAL ESTIMATION OF THE DIFFERENCE BETWEEN TWO POPULATION PARAMETERS

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1 INTERVAL ESTIMATION OF THE DIFFERENCE BETWEEN TWO POPULATION PARAMETERS

2 Estimating the difference of two means: μ 1 μ Suppose there are two population groups: DLSU SHS Grade 11 Male (Group 1) and Female (Group ) students. Variable of Interest: GENMATH Final Grade GOAL: Estimate the difference between the population mean GENMATH grade of DLSU SHS Grade 11 Male and Female students.

3 Estimating the difference of two means: μ 1 μ The point estimator for the difference between the population means μ 1 μ is given by the difference of the sample means: x 1 x.

4 Estimating the difference of two means: μ 1 μ Case 1A: Independent Samples (σ 1 and σ known) A (1 α)(100%) confidence interval estimate for μ 1 μ is [ x 1 x e, x 1 x + e], where σ 1 e = zα + σ if independent samples are taken n 1 n from normal populations with sample sizes n 1 and n and with σ 1 and σ known.

5 Estimating the difference of two means: μ 1 μ Case 1B: Independent Samples (σ 1 and σ unknown and n 1 and n large) A (1 α)(100%) confidence interval estimate for μ 1 μ is [ x 1 x e, x 1 x + e], s 1 where e = zα + s if independent samples n 1 n are taken from nonnormal populations with large sample sizes n 1 and n with σ 1 and σ unknown.

6 Example 11.1: Estimating the difference of two means: μ 1 μ A professor was interested in determining the length of time (in hours h) female and male students spend studying statistics during weeknights. The professor selected a random sample of 81 female and 100 male students, and asked each student to indicate the length of time he/she spends studying statistics on a weeknight. Results are given in the next slide.

7 Example 11.1: Estimating the difference of two means: μ 1 μ

8 Example 11.1: Estimating the difference of two means: μ 1 μ A. Find a point estimate of the difference between the true mean length of time female and male students spend studying statistics on weeknights.

9 Example 11.1: Estimating the difference of two means: μ 1 μ B. Construct a 90% confidence interval for the difference between the true mean length of time female and male students spend studying statistics on weeknights.

10 Estimating the difference of two means: μ 1 μ Case 1C: Independent Samples (σ 1 and σ unknown but assumed equal and n 1, n < 30) A (1 α)(100%) confidence interval estimate for μ 1 μ is [ x 1 x e, x 1 x + e], where e = tα s p 1 n n and s p = n 1 1 s 1 + n 1 s with df = n n 1 +n 1 + n if independent samples are taken from normal populations with small sample sizes n 1 and n.

11 Example 11.: Estimating the difference of two means: μ 1 μ A comparison of the cost of renting an apartment (in thousands of pesos) in two different cities is given below.

12 Example 11.: Estimating the difference of two means: μ 1 μ A. Find a point estimate of the difference between the true mean cost of renting an apartment in the two cities.

13 Example 11.: Estimating the difference of two means: μ 1 μ B. Assume that the cost of renting an apartment in these two cities are approximately normal with unknown but equal population variances. Construct a 90% confidence interval for the difference between the true mean cost of renting an apartment in the two cities.

14 Estimating the difference of two means: μ 1 μ Case 1D: Independent Samples (σ 1 and σ unknown assumed unequal and n 1, n < 30) A (1 α)(100%) confidence interval estimate for μ 1 μ is [ x 1 x e, x 1 x + e], where s 1 e = tα + s if independent samples are taken n 1 n from normal populations with small sample sizes n 1 and n. The degrees of freedom is shown in the next slide.

15 Estimating the difference of two means: μ 1 μ v = s 1 n 1 s 1 n 1 + s n (n 1 1) + s n 1 (n 1) Rounded down to the nearest integer.

16 Example 11.3: Estimating the difference of two means: μ 1 μ A professor teaches two sections of the same course. One meets at 8AM and the other at 10AM. Identical exams are given to both sections. The professor suspects that their scores will differ, on the average, because of a question leakage for the exam. Random sample of students were taken from each section. A summary of the students scores is given in the next slide.

17 Example 11.3: Estimating the difference of two means: μ 1 μ

18 Example 11.3: Estimating the difference of two means: μ 1 μ A. Find a point estimate of the difference between the true mean score of the two sections.

19 Example 11.3: Estimating the difference of two means: μ 1 μ B. Suppose that the scores in the two sections are approximately normal with unknown and assumed unequal population variances. Construct an approximate 99% confidence interval for the difference between the true mean score of the two sections.

20 Estimating the difference of two means: μ 1 μ (Paired) Two sets of samples are called paired or related samples if the two sets were taken from the same population. Example: The population of study is the set of all people enrolled in a weight loss program from a particular gym. The measurements taken from the elements of the sample are their weights before and weights after the program.

21 Estimating the difference of two means: μ 1 μ (Paired) The point estimator for the difference between the population means μ 1 μ is given by the average of the differences of the n pairs of observations d = n i=1 n d i.

22 Estimating the difference of two means: μ 1 μ (Paired) Case : Paired Samples A (1 α)(100%) confidence interval estimate for μ 1 μ is [d e, d + e], where s d n e = tα if related samples of size n have observed differences following the normal distribution with s d as the sample standard deviation of the differences from each pair and df = v = n 1.

23 Example 11.4: Estimating the difference μ 1 μ (Paired) In a study of the effectiveness of physical exercise in weight reduction, a sample of eight persons who engaged in a prescribed program of physical exercise for one month showed the following results:

24 Example 11.4: Estimating the difference μ 1 μ (Paired) A. Find a point estimate of the true mean difference between the weights before and after the physical exercise program.

25 Example 11.4: Estimating the difference μ 1 μ (Paired) B. Construct a 90% confidence interval for the true mean difference between the weights before and after the physical exercise program.

26 Estimating the difference of two proportions: p 1 p Consider a random sample of n 1 taken from population 1 that follows a binomial distribution with an unknown proportion of success p 1. Another independent random sample of size n taken from population that also follows a binomial distribution with an unknown proportion of success p.

27 Estimating the difference of two proportions: p 1 p A point estimator of the difference between population proportions p 1 p is given by p 1 p.

28 Estimating the difference of two proportions: p 1 p A (1 α)(100%) confidence interval estimate for p 1 p is [ p 1 p e, p 1 p + e], where p 1 q 1 n 1 + p q n e = zα sufficiently large. if n 1 and n are

29 Example 11.5: Estimating the difference proportions, p 1 p A school administrator wants to investigate the effect of climate on students attendance in school. Two groups of students were selected at random, one group from the uplands, and the other from the lowlands. Of the 300 students from the uplands, 7 were absent from school for at least one day during the semester, and of the 400 students from lowlands, 70 were absent from school for at least one day.

30 Example 11.5: Estimating the difference proportions, p 1 p A. Find a point estimate of the difference between the true proportion of students from the uplands and lowlands who were absent from school for at least one day during the semester.

31 Example 11.5: Estimating the difference proportions, π 1 π B. Construct a 95% confidence interval for the difference between the true proportion of students from the uplands and lowlands who were absent from school for at least one day during the semester. Interpret this result.

32 Example 11.5: Estimating the difference proportions, π 1 π C. From the results in (B), can you conclude that a colder climate results to a higher proportion of students who are absent from school for at least one day during the semester? Justify.

Estimating a Population Mean

Estimating a Population Mean MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2017 Objectives At the end of this lesson we will be able to: obtain a point estimate for