(Basic Statistics for Citizen Data Scientist) Two Sample t Test: equal variances We now consider an experimental design where we want to determine whether there is a difference between two groups within the population. For example, let’s suppose we want to test whether there is any difference between the effectiveness of a new drug for …

# Month: January 2021

(Basic Statistics for Citizen Data Scientist) One Sample t Test The t distribution provides a good way to perform one-sample tests on the mean when the population variance is not known provided the population is normal or the sample is sufficiently large so that the Central Limit Theorem applies. It turns out that the t distribution provides good results even …

(Basic Statistics for Citizen Data Scientist) Basic Concepts of t Distribution The one sample hypothesis test described in Hypothesis Testing using the Central Limit Theorem using the normal distribution is fine when one knows the standard deviation of the population distribution and the population is either normally distributed or the sample is sufficiently large that the Central Limit …

(Basic Statistics for Citizen Data Scientist) Required Sample Size for the Binomial Testing We now show how to determine the sample size required to achieve a specified power objective. Example 1: A company has made a major improvement in their manufacturing process and wants to test whether this improvement will result in 80% of the components …

(Basic Statistics for Citizen Data Scientist) Statistical Power for the Binomial Distribution Power of one-tailed test Example 1: What is the power of the test in Example 3 of Hypothesis Testing for the Binomial Distribution? For this example we found 13 successes in a sample of size 24 and used a one-tailed test with α = .05 …

(Basic Statistics for Citizen Data Scientist) Runs Example 1: What is the probability that there will be a run of at least 6 heads in 20 tosses of a fair coin? We solve this problem by recursion. Let p = the probability that a heads will occur on any toss, r = the size of run we are looking …

(Basic Statistics for Citizen Data Scientist) Poisson Distribution Basic Concepts Definition 1: The Poisson distribution has a probability distribution function (pdf) given by The parameter μ is often replaced by λ. A chart of the pdf of the Poisson distribution for λ = 3 is shown in Figure 1. Figure 1 – Poisson Distribution Observation: Some key statistical properties of the Poisson …

(Basic Statistics for Citizen Data Scientist) Negative Binomial and Geometric Distributions Negative Binomial Distribution Definition 1: Under the same assumptions as for the binomial distribution, let x be a discrete random variable. The probability density function (pdf) for the negative binomial distribution is the probability of getting x failures before k successes where p = the probability of success on any single trial. Thus …

(Basic Statistics for Citizen Data Scientist) Two-sample Proportion Testing Theorem 1: Let x1 and x2 be random variables with proportional distributions with mean π1 and π2 respectively. Let p1 be the proportion of successes in n1 trials of the first distribution and let p2 be the proportion of successes in n2 trials of the second distribution. When the number of trials n1 and n2 are sufficiently large, usually when ni πi ≥ 5 and ni (1 –πi) ≥ 5, the difference between …

(Basic Statistics for Citizen Data Scientist) One-sample Proportion Testing From the theorem, we know that when sufficiently large samples of size n are taken, the distribution of sample proportions is approximately normal, distributed around the true population proportion mean π, with standard deviation (i.e. the standard error) We can use this fact to do hypothesis testing as was …

(Basic Statistics for Citizen Data Scientist) Hypothesis Testing for Binomial Distribution Example 1: Suppose you have a die and suspect that it is biased towards the number three, and so run an experiment in which you throw the die 10 times and count that the number three comes up 4 times. Determine whether the die is biased. …

(Basic Statistics for Citizen Data Scientist) Binomial Distribution Definition 1: Suppose an experiment has the following characteristics: the experiment consists of n independent trials, each with two mutually exclusive outcomes (success and failure) for each trial the probability of success is p (and so the probability of failure is 1 – p) Each such trial is called a Bernoulli trial. Let x be the discrete …