1. Introduction to Statistics
Elan Ding
Clemson University

2. What is statistics? Statistics answers questions about our world
Data collection
Data summarization (descriptive statistics)
Data analysis (inferential statistics)
Interpretation of results

3. Example
Your friend claimed that among 1000 tosses of a fair coin, she obtained 535 heads.
Is her claim plausible?

4. List of Topics 1. Descriptive Statistics 2. Probability Overview
3. Sampling Distributions
4. Hypothesis Testing
5. Common Statistical Tests

5. 1. Descriptive Statistics
Graphical Methods

6. Perceived Risk in Smoking

7. Comparative Bar Graph

8. Comparative Bar Graph

9. Life Insurance for Cartoon Character

10. Should doctors get auto insurance discount?

11. Stem-leaf plot

12. Math SAT score in 2005

13. Frequency Histogram

14. GPA report errors

15. GPA report errors

16. 1. Descriptive Statistics
Numerical Methods

17. Centrality

18. Variability

19. Quartiles

20. Boxplot

21. NBA Salaries

22. The Central Limit Theorem
2. Probability
The Central Limit Theorem

24. Example
Two players play a game until one player wins two games in a row.
Identify the following:
Experiment
Outcome
Event
Sample Space

25. Probability Distribution

26. Probability Density

27. Probability Density

28. Mean of discrete random variable

29. Mean of continuous random variable

30. Variance of discrete random variable

31. Variance of continuous random variable

32. Binomial Distribution
The probability of k success in n Bernoulli trials is:
What does it look like?
Web app

33. Normal Distribution

34. Standard Normal Distribution

35. Z-Table

36. Standardization

37. The Empirical Rule

38. Example
Suppose 𝑋= the height of a randomly selected 5-year old child follows a normal distribution with 𝜇=100 cm and 𝜎=6 cm.
What proportion of height is between 94 cm and 112 cm?

40. The Central Limit Theorem
Web app

41. 3. Sampling Distribution
The Central Limit Theorem

42. Population vs sample

43. What is a statistic?

44. What is a sampling distribution?
Suppose a random variable 𝑋 has a Bernoulli distribution.
Such that 𝑋=1 with probability 0.4 and 𝑋=0 with probability 0.6.

45. What is a sampling distribution?
Suppose we take a sample of 2 from the population, and call them 𝑋 1 and 𝑋 Define the statistic
𝑇= 𝑋 1 + sin 𝑋 2 .
What is the distribution of 𝑇?
We draw 1000 random samples of size 2 and obtain an approximation:

46. Why sampling distribution?
Populations parameters such as 𝜇 and 𝜎 2 are often unknown.
Sample mean 𝑋 and sample variance 𝑆 2 are called the unbiased estimator for 𝜇 and 𝜎 2 :
𝑋 = 1 𝑛 𝑖=1 𝑛 𝑋 𝑖
𝑆 2 = 1 𝑛−1 𝑖=1 𝑛 ( 𝑋 𝑖 − 𝑋 )

47. Why is sampling distribution useful?
The Central Limit Theorem!
The most important statistic is 𝑋 , which is approximately normal when the sample size is large!
The CLT can be safely applied when 𝑛 exceeds 30.
Web app

48. Example Revisited
Your friend claimed that among 1000 tosses of a fair coin, she obtained 535 heads.
Is her claim plausible?

49. Solution

50. Solution

51. Solution

53. Application of sampling distribution
4. Hypothesis Testing
Application of sampling distribution

54. Forming Hypothesis
Let’s look at the previous problem in a different light.
Suppose now your friend INDEED got 535 heads in 1000 tosses.
Can you say something about whether the coin is fair or not?
To do that we set up the following hypothesis:

55. Forming Hypothesis
Let’s look at the previous problem in a different light.
Suppose now your friend INDEED got 535 heads in 1000 tosses.
Can you say something about whether the coin is fair or not?
To do that we set up the following hypothesis:

56. SUPPOSE 𝐻 0 is true

64. 5. Common Statistical Tests
Application of hypothesis testing

65. 1. Z-test for sample proportion
Researchers at the University of Luton conducted a survey of 321 faculty members at a variety of academic institutions. It was reported that 36% of those surveyed said they occasionally used online searches with key words from student work to check for plagiarism.
Assuming it is reasonable to regard this sample as representative of university faculty members, does the sample provide convincing evidence that more than one-third of faculty members occasionally use key word searches to check student work?

66. 1. Z-test for sample proportion

67. 1. Z-test for sample proportion

68. 1. Z-test for sample proportion

69. 1. Z-test for sample proportion

70. 1. Z-test for sample proportion

71. 𝐻 1 :𝑝≠ 𝑝 0 (two-tailed)

72. 𝐻 1 :𝑝> 𝑝 0 (one-tailed)

73. 2. Z-test for sample mean
A study investigated whether time perception is impaired during nicotine withdrawal.
After a 24-hr smoking abstinence, 20 smokers were asked to estimate how much time had passed during a 45-sec period. Suppose the resulting data on perceived elapsed time (in seconds) were as shown:
Researchers want to know whether smoking abstinence can cause overestimation of elapsed time.

74. 2. Z-test for sample mean

75. 2. Z-test for sample mean

76. The t-distribution

77. t-table

78. Statistical Significance does not imply practical importance

79. 3. t-test for two sample means (paired)

80. 3. t-test for two sample means (paired)

81. 4. t-test for two sample means (unpaired)

82. 4. t-test for two sample means (unpaired)

83. Thank you!