Hypothesis Testing with One Sample Flashcards
Alpha (Type I error)
The probability of rejecting the null hypothesis when the null hypothesis is true
Alternative Hypothese
A claim about the underlying population that contradicts the null hypothesis. This is normally what we are trying to prove.
Notation: Not equal to, greater than, or less than - Never equal to
Beta (Type II error)
The proability fo failing to reject the null hypothesis when the null hypothesis is false
Binomial Distribution
A discrete random variable that arises from Bernoulli trials. There are a fixed number, n, trials and the trials are independent.
Central Limit Theorem
If the size of a sample is sufficiently large, then the distribution of the sample means and distribution of the sums are approximately normal regardless of the shape of the population.
Confidence Interval
An interval estimate for an unknown population parameter. This depends on:
1. The desired confidence level.
2. Information that is known about the distribution (for example, known standard deviation).
3. The sample and its size.
Hypothesis
A statement about the value of a population parameter, in case of two hypotheses,
1. Null hypothesis (notation Ho) - statement assumed to be true
2. Alternative hypothesis (Ha) - statement we accept is the data does not support the null hypothesis.
Hypothesis Decision
- Reject Ho - The results of the sample are significant. Sufficient evidence to conclude Ho is incorect and Ha may be correct
- Do not reject Ho - The sample results are not significant. Insufficient evidence to reject Ho.
Hypothesis Testing
Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion.
1. Set up two contradictory hypotheses.
2. Collect sample data
3. Determine the correct distribution to perform the hypothesis test.
4. Analyze sample data that allow you to reject or decline to reject the null hypothesis.
5. Make a decision and write a meaningful conclusion.
Level of Significance of the Test
Probability of a Type I error (reject the null hypothesis when it is true).
1. Notation: α.
2. In hypothesis testing, the Level of Significance is called the preconceived α or the preset α.
Normal Distribution
a continuous random variable (RV) where μ is the mean of the distribution, and σ is the standard deviation,
1. Notation: X ~ N(μ, σ).
2. Standard Normal distribution: X~N(0,1)
Null Hypothesis
A statement of no difference between the variables; they are not related. This is the status quo.
Notation: Equal to, greater than or equal to, or less than or equal to - always has an equal sign
Power of the hypothesis Test
- 1- beta (1 - the probability of tpe II error)
- Typically want a power close to 1.
- Increasing the sample size increases the power of the test.
Probabilty Distributions for Hypothesis Testing
- t-test (unknown standard deviation) - the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with an unknown standard deviation.
- normal (known standard deviation) - the data come from a simple, random sample and the population is approximately normally distributed, or the sample size is large, with a known standard deviation.
p-value
the probability that an event will happen purely by chance assuming the null hypothesis is true. The smaller the p-value, the stronger the evidence is against the null hypothesis.
Standard Deviation
- Measures how far data values are from their mean
- A number that is equal to the square root of the variance and ;
- Notation: s for sample standard deviation and σ for population standard deviation.
Student’s t-Distribution
The major characteristics of the random variable (RV) are:
1. It is continuous and assumes any real values.
2. The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
3. It approaches the standard normal distribution as n gets larger.
4. There is a “family” of t distributions: every representative of the family is completely defined by the number of degrees of freedom which is one less than the number of data items.
Type I Error
The decision is to reject the null hypothesis when, in fact, the null hypothesis is true.
Type II Error
The decision is not to reject the null hypothesis when, in fact, the null hypothesis is false.
