Statistics chapter 4-6 Alexia Bowen
How do discrete and continuous random variables differ?
A discrete random variable can assume a countable number of values, while a continuous random variable can assume values corresponding to any of the points contained in an interval.
What is a random variable?
A random variable is a variable that assumes numerical values associated with the random outcomes of an experiment, where one (and only one) numerical value is assigned to each sample point
Consider a population that contains values of x equal to 0, 1, 2,..., 97, 98, 99. Assume that the values of x are equally likely. For the sample sizes n=2, n=5, n=10, n=30, and n=50, a computer was used to generate 500 random samples and calculate x overbar for each sample. Relative frequency histograms of the 500 values of x overbar were constructed for each of the sample sizes. What changes occur in the histograms as the value of n increases? What similarities exist? A) What changes occur in the histograms as the value of n increases? B) What similarities exist?
A) As the value of n increases, the histograms become less spread out B) All of the histograms have an approximately normal distribution shape and similar central tendencies.
What are the properties of an ideal estimator?
An ideal estimator is unbiased and has a small variance
Will the sampling distribution of x overbar always be approximately normally distributed? Explain
No, because the Central Limit Theorem states that the sampling distribution of x overbar is approximately normally distributed only if the sample size is large enough
Describe the shape of a normal probability distribution
The distribution is roughly bell shaped
What is a sampling distribution of a sample statistic?
The sampling distribution of a sample statistic is the probability distribution of that statistic.
The expected value of a discrete random variable must be one of the values in which the random variable can result.
false
When estimating the population mean, the sample mean is always a better estimate than the sample median.
false
State the Central Limit Theorem.
in notes
What is the name given to a normal distribution when μ=0 and σ=1?
standard normal
If x overbar is a good estimator for μ, then we expect the values of x overbar to cluster around μ
true
In most situations, the true mean and standard deviation are unknown quantities that have to be estimated.
true
Sample statistics are random variables, because different samples can lead to different values of the sample statistics.
true
The sample mean, x overbar, is a statistic
true
Is the expected value of the probability distribution of a random variable always one of the possible values of x? Explain.
No, because the expected value may not be a possible value of x for one trial, but it represents the average value of x over a large number of trials.