4.1. Introduction…………………………………………………….143
4.2. Areas under continuous probability density functions…………144
Exercises……………………………………………………………145
4.3. The normal distribution………………………………………...147
4.3.1. Cumulative distribution function of the normal distribution…150
4.4. The standard normal distribution……………………………….151
Exercises…………………………………………………………….154
4.5. Standardizing a normal distribution…………………………….155
Exercises…………………………………………………………….157
4.6. The normal distribution approximation to the binomial
distribution………………………………………………….160
Exercises…………………………………………………………….164
4.7. The exponential probability distribution……………………….167
Exercises…………………………………………………………….170
Chapter 5. Sampling distributions………………………………..172
5.1. Sampling and sampling distributions…………………………...172
5.1.1.
Mean and standard deviation of
…………………………..176
5.1.2. Central limit theorem…………………………………………179
Exercises…………………………………………………………….183
5.2. Sampling distribution of a sample proportion………………….186
5.2.1. Population and sample proportions…………………………...186
5.2.2.
Sampling distribution of
.
Its mean and standard deviation..187
5.2.3. Form of the sampling distribution of ……………………...188
Exercises…………………………………………………………….191
5.2. Sampling distribution of a sample variance…………………….194
Exercises…………………………………………………………….197
Chapter 6. Interval estimation………………………………199
6.1. Introduction……………………………………………………..199
6.2. Confidence interval and confidence level………………………200
6.3. Confidence intervals for the mean of population that is………..203
normally distributed: population variance known
Exercises……………………………………………………………..204
6.4. Confidence intervals for the mean of population that is
normally distributed: large sample size……………………..205
Exercises……………………………………………………………..207
6.5. Confidence intervals for the mean of a normal distribution:
population variance unknown: small sample size…………..209
6.5.1. Student’s t distribution………………………………………..209
6.5.2.
Confidence interval for
:
small samples…………………….212
Exercises…………………………………………………………….214
6.6. Confidence intervals for population proportion: Large samples..215
Exercises……………………………………………………………..218
6.7. Confidence intervals for the difference between means of two
normal populations…………………………………………220
6.7.1. Confidence intervals for the difference between means:
paired samples……………………………………………………….220
Exercises……………………………………………………………..222
6.7.2. Confidence intervals for the difference between means
of two normal populations with known variances………….224
Exercises…………………………………………………………….227
6. 8. Confidence interval for the difference between the population
means: unknown population variances that are assumed to be equal.229
Exercises…………………………………………………………….231
6. 9. Confidence interval for the difference between the population
proportions: (large samples)………………………………..233
Exercises…………………………………………………………….235
6. 10. Confidence interval for the variance of a normal distribution..237
Exercises…………………………………………………………….240
6.11. Sample size determination…………………………………….241
6.11.1. Sample size determination for the estimation of mean……...241
6.11.2. Sample size determination for the estimation of proportion...243
Exercises…………………………………………………………….244
Appendix………………………………………………………… ....246
References………………………………………………………….. 253
To the students
Nowadays, all companies use statistical methods in making decisions. Consequently, the study of statistical methods has taken on a prominent role in the education of student majoring in management and economics. Here is some advice that will help you to succeed in statistics (and in other subjects too).
Tip1: Understanding the process of solving a particular type of problem is emphasized on memorizing formulas. In most cases, if you understand the concepts, memorizing a formula becomes completely unnecessary, because you construct the necessary tools when needed.
Tip2: Classes are held for your benefit. If attending class was not important, all university courses would be by correspondence, and your tuition would be much lower. During class your instructor will go over examples, which are important, and most likely not in the book. Statistics courses are sequential, so the stuff you see in, for example lecture 6, will enable you to make sense of a much material you will see in lecture 7. As instructors, we note a definite correlation between grades and class attendance. Go to class!!
Tip3: Statistics books are not meant to be read like novels (even though they are often exciting). It is better generally to read the sections of the book to be covered in lecture through quickly to get some idea of what there is before going to lecture. After the lecture read it through carefully, with pencil and paper in hand, working through examples.
Tip4: Just as you must play a lot of football to be good at it, you must do a lot of statistics problems in order to be successful as well. At minimum, work on every problem your instructor suggests. If you are having trouble or want more practice, work on other problems in that section or get another book and work problems out of it. If you are having trouble getting a correct answer to a problem, think about what is going wrong. By doing these you can learn something new and prevent yourself from making the same error in the future. Work on problems more than once. Work on problems until you can do them quickly. Remember, the process is usually more important than the result.
Tip5: The fastest way to get into trouble in statistics is to not do homework. Remember, similar problems will probably show up on quizzes and exams, where you will be expected to work them quickly and accurately, probably without the book in front of you.
Tip6: Contrary to many students’ opinions, your instructor wants you to succeed. Extremely rare is the instructor who will intentionally put completely different material on an exam that was covered in class. For this reason, pay attention to your instructor and take notes. Then read your notes, and be sure you understand them, filling any missing details. Review your notes regularly.
The goal of this book is to present statistics in a clear and interesting way. Students, who will use this book, are not required to have a strong background in mathematics. The chapters are divided into sections, and each section contains necessary theoretical background and solved problems. A set of exercises appears at the end of each section. Answers are right after exercises.
In the end, any suggestion from readers would be greatly appreciated.
Dr. Humbet Aliyev