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Remember, the ____ of statistical inference is to use information obtained from a ______ and __________ the results to the population being studied. As with estimating the population proportion, the first step is to obtain a _____ ________ of the parameter.The _____ ________ of the population mean, μ, is the sample mean, x¯.
goal, sample, generalize, point estimate, point estimate
Suppose that a simple random sample of size n is taken from a population. If the population from which the sample is drawn follows a normal distribution, the distribution of
Student’s t-distribution with n−1 degrees of freedom, where x¯ is the sample mean and s is the sample standard deviation.
The six steps of Properties of the t-Distribution are:
1) The t-distribution is different for different degrees of freedom.
2) The t-distribution is centered at 0 and is symmetric about 0.
3) The area under the curve is 1. The area under the curve to the right of 0 equals the area under the curve to the left of 0, which equals 1/2.
4) As t increases or decreases without bound, the graph approaches, but never equals, zero.
5) The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution because we are using s as an estimate of σ, thereby introducing further variability into the t-statistic.
6) As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because as the sample size increases, the values of s get closer to the value of σ by the Law of Large Numbers.
is used to represent the z-score whose area under the normal curve to the right of
. Similarly, we let
represent the t-value whose area under the t-distribution to the right of
The shape of the t-distribution depends on the sample size, n. Therefore, the value of tα depends not only on α, but also on the degrees of freedom, n−1.
List the three conditions required for constructing a confidence interval for a population mean
sample data come from a simple random sample or randomized experiment,
sample size is small relative to the population size (n≤0.05N),
the data come from a population that is normally distributed, or the sample size is large
List two things you should look for that tell you to construct a confidence interval for a population proportion,
1) sample data that are qualitative with 2 outcomes (e.g., true/false, yes/no, 0 or 1)
2) summary information that includes a sample proportion - the number of successes x out of n trials.
List the three conditions that must be met when constructing a confidence interval for a population proportion,
1) the sample is obtained by simple random sampling or through a randomized experiment.
3) the sample size can be no more than 5% of the population size
list two things you should look for that tell you to construct a confidence interval for a population mean,
1) raw data that are quantitative,
2) summary information that includes a sample mean (
), sample standard deviation (
), and sample size (
Besides the facts that...
(1) the sample must be obtained by simple random sampling or through a randomized experiment and
(2) that the sample size must be small relative to the size of the population,
what other condition must be met?
The data come from a population that is at least approximately normal with no outliers.
This condition is verified using a normal probability plot (to check for normality) and a boxplot (to check for outliers).
See below for a table of
The steps listed in the flowchart are as follows:
(1) Which parameter are we estimating?
If “Proportion, p,”
and construct a confidence interval for p.
If “mean, μ,”
forward to "Is n ≥ 30?"
(2) Is n ≥ 30?
If “Yes”, forward to "Compute t-interval"
If “No”, "Do the data come from a population that is approximately normal with no outliers?"
(3) Do the data come from a population that is approximately normal with no outliers?
If “Yes”, Compute t-interval
If “No”, Use nonparametric methods
9 - Video
In general, how are data used to test hypotheses?
If the data are unusual, the null hypothesis is rejected.