ST350 Test 2
Determining the sample size
n= [(za/2σ)/E]^2. For a proportion it is n= [z(a/2)/E]^2 * p(1-p).
Hypothesis testing process
- State the null and alternative hypotheses (also decide if mean or proportion) - define a test statistic - do the computations - Asses the evidence - state conclusion in terms of the original problem/question.
To calculate normal probabilities
- sketch the normal curve - shade the region of interest and mark the delimiting x values - compute the Z-scores for the delimiting X-values - use the standard normal table to obtain the probabilities.
Sampling distribution of the mean
For a variable X and a sample size n, the distribution of the variable (line over X) is called the sampling distribution of the mean. As the sample size increases, the sampling error decreases. For a sample size n, the mean of (line over X)= the mean of the variable under consideration. For a sample size n, the standard deviation of (line over X)= the standard deviation of X/the square root of the sample size n.
Central limit theorem
If X is a normally distributed random variable with a mean μ and standard deviation σ, then the sample mean (line over X) has a normal distribution with mean μ and standard distribution σ/ square root of n.
To find the Z-score
If you are given a normal probability - sketch the normal curve - shade the region of interest - use the table to get the z-score - convert the z-score to an x-value using the formula X= μ + σz
Point estimate for a proportion
Might need to know the proportion of times a particular outcome occurs. p=population proportion of an event. For a sample size n if the sample is no larger than 10% of the population and the sample size is large enough that np > or equal to 10 AND np(1-p) > or equal to 10, then the sample p has a normal distribution with a mean p and standard deviation square root[(p(1-p))/n]. A (1-a)% confidence interval is given by the formula on the formula sheet. It has its own table (:
Null hypothesis v. alternative hypothesis.
Null- The hypothesis we assume to be true. Represented by Ho. μ=μ(subscript o) for the mean. p = p(subscript o) for the proportion. Alternative- The hypothesis we put forward as an alternative to the null. What we want to prove. Represented by Ha. It's form varied based on the hypothesis proposed. - If deciding whether the mean is different from a specified value, μ =(w slash through) μ(subscript o) [2 sided or 2 tailed] repeat for p - If deciding if the mean is less than/greater than something μ >/< μ(subscript o) [1 sided or 1 tailed] repeat for p
Sampling error
The distribution of a sample statistic is called the sampling distribution. Sampling error is the error resulting from using a sample statistic to estimate a population parameter. This helps us understand how accurate an estimate is likely to be.
Normal distribution
The most important of continuous probability distributions. Standard curve peaking in middle. If all possible values of X follow an assumed normal curve, then X is said to be a normal random variable, and the population is normally distributed. The normal distribution can be defined by the mean (μ) and standard deviation (σ). They describe the population and can be estimated by the sample mean and sample standard deviation. A standard normal distribution has the mean 0 and a standard deviation of 1. To standardize a variable X : Z= (X-μ)/σ
Point estimate
The value of a single statistic used to estimate the population parameter. Line over x (sample mean) is a good point estimate for the population mean.
Continuous random variable
A variable that can assume any value on a continuum (can assume an uncountable number of values). Examples: thickness of an item, time required to complete a task, temperature of a solution, height in inches. These can potentially take on any value, depending on the ability to measure accurately. If a random variable is continuous, its distribution is called a continuous probability distribution. This is different from a discrete probability distribution because: - the probability that a continuous random variable will assume a particular value is 0. - an equation or formula is used to describe a continuous probability distribution. - the area under the graph is 1. - the probability that a random variable assumes a value between a and b is equal to the area under the curve between a and b.
Confidence interval
Consists of a point estimate and a margin of error along with a specified confidence level for the interval. μ doesn't change, the values of the sample means and the endpoints of the intervals change. Confidence intervals enable you to say: I am 95% confident that this interval contains the true population mean. If you know the standard deviation: a 100*(1-a)% confidence interval for the mean of a normal population with a known standard deviation is given by: (line over x) + or - Z (subscript a/2) * (σ/ square root of n). The higher the confidence interval, the wider the confidence interval. The higher the sample size, the smaller the confidence in the interval. a is based off of the confidence interval specified for you. a/2 is a critical value from the table. If you don't know the standard deviation: You have to use the sample data to estimate σ... use sample standard deviation (s). This creates a t-distribution. This looks a lot like a normal distribution, but it shape depends on the sample size based on a quantity called the degrees of freedom(df).. (n-1). It basically states that for any given sample mean only n-1 of the sample values are free to vary while estimating the mean. As n increases, the t-distribution looks more like a normal distribution. It has its own table! A 100*(1-a) confidence interval for the mean of a normal population where σ is not known is given by line over X + or - t(subscript a/2) * (s/square root of n). The formula also applies for non-normal data with a large sample size (n>30).
