CHAPTER 4 PART 2 STATISTICAL INFERENCES 1 Confidence

Confidence interval for population proportion Confidence Interval for a Proportions Confidence Interval for Difference

4. 2. 3 Confidence Interval Estimates for Population Proportion 3

EXAMPLE 4. 4 A survey was conducted to estimate the proportion of the households

95% confidence interval 7 Thus, we can state with 95% confidence that proportion of

EXAMPLE 4. 5 229 students answered the question in the class questionnaire about whether

EXERCISE 1 A random sample of 400 electronic components manufactured by a certain process

EXAMPLE 4. 6 In a survey of 100 people from the city A, 55

SOLUTION City A : n 1 = 100 and x 1 = 55 City

For 99% confidence interval, Zα/2 = Z 0. 01/2 = Z 0. 005 =

EXAMPLE 4. 7 In a study to compare the effects of two pain relievers,

EXERCISE 1 Samples of 400 printed circuit boards were selected form each of two

4. 2. 5 Error of Estimation and Determining the Sample size Definition 4. 3:

EXAMPLE 4. 8 The dean of a university wants to use the mean of

SOLUTION Given 95% confidence interval, Zα/2 = 1. 96 E = 0. 25, σ

$EXAMPLE 4. 7 A quality control engineer wants to estimate the fraction of defectives$

SOLUTION Given 95% confidence interval, Zα/2 = 1. 96 E = 0. 01, Use

EXAMPLE 4. 8 A researcher wishes to estimate, with 95% confidence, the proportion of

EXTRA EXERCISES The mean and standard deviation of the speeds of the sample of

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CHAPTER 4 (PART 2) STATISTICAL INFERENCES 1

Confidence interval for population proportion Confidence Interval for a Proportions Confidence Interval for Difference Between Two Proportions Sample Size Calculation 2

4. 2. 3 Confidence Interval Estimates for Population Proportion 3

EXAMPLE 4. 4 A survey was conducted to estimate the proportion of the households with a personal computer. On of the 300 households surveyed, 75 had a personal computer. Find a 95% confidence interval for the proportion in the population who has a personal computer. 5

95% confidence interval 7 Thus, we can state with 95% confidence that proportion of all households with a personal computer is between 20. 1% and 29. 9%.

EXAMPLE 4. 5 229 students answered the question in the class questionnaire about whether or not they support the building of the new Anali stadium. Of these 229, 167 said they did support the building of new stadium. Estimate the proportion of students who support the building of the stadium using 99% confidence interval. 8

EXERCISE 1 A random sample of 400 electronic components manufactured by a certain process are tested, and 30 are found to be defective. Let p represent the proportion of components manufactured by this process that are defective. Find a 95% confidence interval for p. 11

EXAMPLE 4. 6 In a survey of 100 people from the city A, 55 people preferred reading Time Post. Meanwhile 46 from 150 people preferred reading Time Post in city B. Find a 99% confidence interval estimate for the difference in the two proportions. 1

SOLUTION City A : n 1 = 100 and x 1 = 55 City B : n 2 = 150 and x 2 = 46 The sample proportions are; 1

For 99% confidence interval, Zα/2 = Z 0. 01/2 = Z 0. 005 = 2. 576 15 Since 0 is not included in the interval, we can state with 99% confidence that there is a difference in the two proportions for people who preferred reading Time post.

EXAMPLE 4. 7 In a study to compare the effects of two pain relievers, it was found that of n 1=200 randomly selected individuals instructed to use Excedrin, 93% indicated that it relieves their pain. Of n 2=450 randomly selected individuals instructed to use the Tylenol, 96% indicated that it relieves their pain. Find a 99% confidence interval for the difference in the proportions experiencing relief from pain for these two pain relievers. 1

17 SOLUTION

EXERCISE 1 Samples of 400 printed circuit boards were selected form each of two production lines A and B. Line A produced 40 defectives, and line B produced 80 defectives. Estimate the difference in the actual fractions of defectives for the two lines with a confidence coefficient of 0. 9. 19

4. 2. 5 Error of Estimation and Determining the Sample size Definition 4. 3: 20

Definition 4. 4: 21

EXAMPLE 4. 8 The dean of a university wants to use the mean of a random sample to estimate the average amount of time taken by the students to get from one class to the next class. She wants to assert with probability 0. 95 that her error will be at most 0. 25 minute. If she knows from the previous studies, minutes, how large a sample will she need? 2

SOLUTION Given 95% confidence interval, Zα/2 = 1. 96 E = 0. 25, σ = 1. 5 Use formula 2

$EXAMPLE 4. 7 A quality control engineer wants to estimate the fraction of defectives$

EXAMPLE 4. 7 A quality control engineer wants to estimate the fraction of defectives in a large lot of film cartridges. From the previous experience, he feels that the actual fraction of defectives should be somewhere around 0. 05. How large a sample should he take if he wants to estimate the true fraction to within 0. 01, using a 95% confidence interval? 2

SOLUTION Given 95% confidence interval, Zα/2 = 1. 96 E = 0. 01, Use formula 2

EXAMPLE 4. 8 A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher took a sample size n = 2305. Calculate the error of estimation. 2

SOLUTION 2

EXTRA EXERCISES The mean and standard deviation of the speeds of the sample of 69 skaters at the end of the 6 meter distance were 5. 753 and 0. 892 meters per second, respectively. a) Find a 95% confidence interval for the mean velocity at the 6 -meter mark. Interpret the interval. b) Suppose you wanted to repeat the experiment and you wanted to estimate this mean velocity correct to within 0. 01 second, with probability 0. 99. How many skaters would have to be included in your sample? 28

EXTRA EXERCISES (from Exercise 1) • 29