Introduction to Statistics Elan Ding Clemson University

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

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

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

1. Descriptive Statistics Graphical Methods

Perceived Risk in Smoking

Comparative Bar Graph

Comparative Bar Graph

Life Insurance for Cartoon Character

Should doctors get auto insurance discount?

Stem-leaf plot

Math SAT score in 2005

Frequency Histogram

GPA report errors

GPA report errors

1. Descriptive Statistics Numerical Methods

Centrality

Variability

Quartiles

Boxplot

2003-2004 NBA Salaries

The Central Limit Theorem 2. Probability The Central Limit Theorem

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

Probability Distribution

Probability Density

Probability Density

Mean of discrete random variable

Mean of continuous random variable

Variance of discrete random variable

Variance of continuous random variable

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

Normal Distribution

Standard Normal Distribution

Z-Table

Standardization

The Empirical Rule

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?

The Central Limit Theorem Web app

3. Sampling Distribution The Central Limit Theorem

Population vs sample

What is a statistic?

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.

What is a sampling distribution? Suppose we take a sample of 2 from the population, and call them 𝑋 1 and 𝑋 2 . 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:

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 )

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

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

Solution

Solution

Solution

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

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:

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:

SUPPOSE 𝐻 0 is true

5. Common Statistical Tests Application of hypothesis testing

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?

1. Z-test for sample proportion

1. Z-test for sample proportion

1. Z-test for sample proportion

1. Z-test for sample proportion

1. Z-test for sample proportion

𝐻 1 :𝑝≠ 𝑝 0 (two-tailed)

𝐻 1 :𝑝> 𝑝 0 (one-tailed)

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.

2. Z-test for sample mean

2. Z-test for sample mean

The t-distribution

t-table

Statistical Significance does not imply practical importance

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

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

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

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

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