g treated with treatment A or with treatment B. From the study results, we want to draw a conclusion that goes beyond the study population-is treatment A more effective than treatment B in the total universe of all patients with this disease who might be treated with treatment A or treatment B?
Under the assumption that the binomial distribution will approximate the normal distribution The probability of at least 52 of 100 attempts 0.3821 (high probability to be due to chance)
In probability and statistics, Student’s t-distribution (or simply the t-distribution) is a continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown. It plays a role in a number of widely-used statistical analyses, including the Student’s t-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student’s t-distribution also arises in the Bayesian analysis of data from a normal family. The t-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student’s t-distribution is a special case of the generalized hyperbolic distribution.
Binomial Experiment A binomial experiment (also known as a Bernoulli trial) is a statistical experiment that has the following properties: ■ The experiment consists of n repeated trials. ■ Each trial can result in just two possible outcomes. We call one of these outcomes a success and the other, a failure. ■ The probability of success, denoted by P, is the same on every trial. ■ The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials. Consider the following statistical experiment. You flip a coin 2 times and count the number of times the coin lands on heads. This is a binomial experiment because: ■ The experiment consists of repeated trials. We flip a coin 2 times. ■ Each trial can result in just two possible outcomes - heads or tails. ■ The probability of success is constant - 0.5 on every trial. ■ The trials are independent; that is, getting heads on one trial does not affect whether we get heads on other trials. Notation The following notation is helpful, when we talk about binomial probability. ■ x: The number of successes that result from the binomial experiment. ■ n: The number of trials in the binomial experiment. ■ P: The probability of success on an individual trial. ■ Q: The probability of failure on an individual trial. (This is equal to 1 - P.) ■ b(x; n, P): Binomial probability - the probability that an n-trial binomial experiment results in exactly x successes, when the probability of success on an individual trial is P. ■ nCr: The number of combinations of n things, taken r at a time. Binomial Distribution A binomial random variable is the number of successes x in n repeated trials of a binomial experiment. The probability distribution of a binomial random variable is called a binomial distribution (also known as a Bernoulli distribution). Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below.
A large z value would be the one above 2 02 below -2
Z = + or – 1.96 is equal to an alpha level of .05
Alpha is divided between the two tails 0.05 .025 each
Alpha is one side 0.05
Alpha is in one side 0.05
Z = + or – 1.96 is equal to an alpha level of .05
Given this background, let us now consider a trial in which groups receiving one of two 