STAT121 Exam 3
Fill in the blank: The advantage of _____ over _____ is to remove variation associated with the blocking variable from experimental error.
"randomized block experiment" over "completely randomized experiment"
What is possibly a significant result due to the very large sample size when there should not be significance?
A test of significance on a very large random sample that has very little chance variation.
Fill in the blank: If x-bar is within margin of error of μ, then μ will be within _____ of x-bar.
Margin of error
What is in the planning stage?
The name for how often the confidence interval estimation procedure produces 98% confidence intervals that contain the value of the parameter being estimated.
Central Limit Theorem
The name of the statement telling us that when sampling from a non-Normal population, the sampling distribution of ¯x is approximately Normal whenever the sample is large and random.
Claimed parameter value
The value of the parameter given in the null hypothesis. E.g., μ0 is a claimed parameter value.
Sampling variability
The variability of sample results from one sample to the next—something we must measure in order to effectively do inference. Margin of error only covers sampling variability.
What is using the wrong formula?
using the formula x-bar + or - t*(s divided by square root n) for data from a stratified sample.
Standard deviation (s)
A measure of the variability (spread) of data in a sample
Decreased
What happens to the width of a confidence interval when sample size is increased (or level of confidence is decreased.)
Standard deviation of x-bar
A measure of the variability of the sampling distribution of x-bar; equals σ divided by square root n .
Standard error of x-bar
A measure of the variability of the sampling distribution of x-bar ; estimates the standard deviation of the sampling distribution of x-bar ; computed using the formula: s divided by square root n.
Parameter
A characteristic of a population that is usually unknown; this characteristic could be the mean (μ), median, proportion, standard deviation (σ), etc.
Statistic
A characteristic of a sample; a number computed from sample data (without any knowledge of the value of a parameter) used to estimate the value of a parameter. Examples include x , the sample mean, and s, the sample standard deviation.
Degrees of freedom
A characteristic of the t-distribution (e.g., n - 1 for a one-sample t); a measure of the amount of information available for estimating σ using s.
Equal variance
A condition for ANOVA; the condition is met when the largest standard deviation divided by the smallest standard deviation is less than 2.
Unbiased
A condition where the mean of all possible statistic values equals the parameter that the statistic estimates.
Statistical significance
A difference between the observed statistic and the claimed parameter value as given in H0 that is too large to be due to chance. (An observed effect that is too large to be due to chance.) Results are significant when P-value < α. Just because results are statistically significant does not imply that the results are important.
Practical significance
A difference between the observed statistic and the claimed parameter value that is large enough to be worth reporting. To assess practical significance, look at the numerator of the test statistic and ask 'Is it worth anything?' If yes, then results are also of practical significance. Note: Do not assess practical significance unless results are statistically significant.
t distribution
A distribution specified by degrees of freedom used to model test statistics for the one sample t test, the two-sample t test, etc. where σ ('s) is (are) unknown. Also used to obtain a confidence interval for estimating a population mean, or the difference between two population means, etc.
What is sampling distribution of x-bar centered at μ0.
A list of the possible values of x-bar is H0: μ = μ0 is true.
Test statistic
A number that summarizes the data for a test of significance and is used to obtain a P-value.
ANOVA (Analysis of Variance)
A statistical procedure for testing the equality of means using variances.
Robust
A statistical procedure that is not sensitive to moderate deviations from an assumption upon which it is based; in other words, the confidence level or P-value does not change very much if the conditions for use are not met. e. g., t procedures give P-values and confidence levels that are very close to correct even when data are not Normally distributed provided the data are not strongly skewed and include no outliers.
Two-sample t procedure
A statistical procedure used to compare the means from two populations either with a test of their equality or by estimating the difference between the two population means.
Conservative two-sample t test
A test for comparing the means from two independent samples or two treatments where the degrees of freedom are taken to be the minimum of (n1 - 1) and (n2 - 1).
Approximate two-sample t test
A test for comparing the means of two independent samples or two treatments where the test statistic has an approximate t distribution. The formula for computing degrees of freedom is complicated.
Pooled two-sample t test
A test for comparing the means of two independent samples or two treatments where the test statistic has an exact t distribution. Degrees of freedom = n1 + n2 - 2. Because this test requires that the two populations have equal variances, the approximate two-sample t is recommended.
What is possibly an insignificant result due to the small sample size when there should be statistical significance?
A test of significance on a small random sample that has a lot of chance variation.
Two-sided test (also called a two-tailed test)
A test where the alternative hypothesis contains "≠".
One-sided or one-tailed test
A test where the alternative hypothesis contains either "<" or ">"
Lower tailed test (also called a left-tailed test)
A test with "<" in the alternative hypothesis. This is a one-sided test.
Upper-tailed test (also called a right-sided test)
A test with ">" in the alternative hypothesis.
Standard error of a statistic
An estimate of the standard deviation of the sampling distribution of the statistic; in other words, it is a measure of the variability of the statistic. Note: The denominators of most test statistics are called standard errors.
Confidence interval
An estimate of the value of a parameter in interval form with an associated level of confidence; in other words, a list of reasonable or plausible values for the parameter based on the value of a statistic. E.g., a confidence interval for μ gives a list of possible values that μ could be based on the sample mean.
One-sample t test
An inferential statistical procedure that uses the mean from one sample of data for either estimating the mean of the population or testing whether the mean of the population equals some claimed value.
Outlier
An observation that falls outside the pattern of the data set. For one sample of data, an outlier will be any observation that is a long way from the rest of the data.
What is failing to reject the null hypothesis since zero is contained in the interval?
Appropriate statistical conclusion when using the 95% confidence interval for μ1 - μ2, namely using the interval (-2.23, 1, 17) to test H0: μ1 - μ2 = 0.
Matched pairs
Either two measurements are taken on each individual such as pre and post OR two individuals are matched by a third variable (different from the explanatory variable and the response variable) such as identical twins or windows matched by installer when comparing installation time of two brands of windows.
What is confidence interval?
Its purpose is to give a range of plausible values for the population parameter.
What is level of confidence or 98%?
Its purpose is to give a range of plausible values for the population parameter.
What is making a distinction when there is no practical distinction?
Making a big deal about a P-value of 0.049 and declaring a P-value of 0.051 to be not significant.
What is multiple analyses?
More than one statistical analysis performed on a data set.
Conditions necessary for ANOVA
Normality of all populations, equality of variances & either stratified sample (independent SRS's) or random allocation. Check (1) data collection (2) if n1 + n2 + . . . + nk < 40, check for outliers in all k data plots; if n1 + n2 + . . . + nk ≥ 40, apply CLThm and (3) largest standard deviation divided by smallest standard deviation < 2.
Conditions necessary for a two-sample t procedure (using t* for C.I. or getting P-value from t table)
Normality of both populations & either stratified sample (independent SRS's) or random allocation. Check (1) data collection and (2) if n1 + n2 < 40, check for outliers in both data plots; if n1 + n2 ≥ 40, apply CLThm.
Conditions necessary for matched pairs t procedure (using t* for C.I. or getting P-value from t table)
Normality of population of differences & either SRS or random allocation. Check (1) data collection and (2) if number of pairs < 40, check for outliers in plot of differences; if number of pairs ≥ 40, apply CLThm.
Conditions necessary for a one-sample t procedure (using t* for C.I. or getting P-value from t table)
Normality of the original population & SRS. (Note: a t-distribution is robust with respect to non- normality provided no outliers and no strong skewness. So, we can use a t-distribution procedure when n < 40 provided the data have no outliers. We must have an SRS however.) Check (1) data collection and (2) if n < 40, check for outliers in data plot; if n ≥ 40, apply CLThm.
Fill in the blank: Hypotheses are always statements about _____.
Parameters or parameter values
Multiple analyses
Performing two or more tests of significance on the same data set. This inflates the overall α (probability of a type I error) for the tests. (The more analyses performed, the greater the chance of falsely rejecting at least one true null hypothesis.)
What is a two-sample t procedure?
Procedure for analyzing data where the explanatory variable is categorical with only two categories and the response variable is quantitative.
What is ANOVA?
Procedure for analyzing data where the explanatory variable is categorical with three or more categories and the response variable is quantitative.
Test of significance
Procedure used to assess the evidence against a claim (hypothesis) about the value of a parameter.
What are the two appropriate methods of data collection for valid inference?
Random allocation of individuals to treatments or random selection of individuals from independent populations.
Garbage
Results from statistical analyses performed on non-random samples or experimental data obtained without random allocation of treatments to individuals.
What is not practically significant?
Results of a significant test of hypotheses where the difference is not large enough to be important or of worth.
What is statistically significant?
Results of a study that differ too much from what we expect due to just randomization to attribute to chance.
Statistically significant
Results of a study that differ too much from what we expected because of randomization to attribute to chance variation.
What are results that may be worthless or meaningless?
Results of a test where theh data were not appropriately collected through probability sampling or randomization.
Reject H0
The appropriate statistical conclusion when P-value < α.
Fail to reject H0
The appropriate statistical conclusion when the P-value is greater than α.
Conditions
The basic premises for inferential procedures. If the conditions are not met, the results may not be valid.
Laws of probability
The basis for hypothesis testing and confidence interval estimation.
What is one?
The biggest value that P-value could be.
What is increased?
The change in the width of a confidence interval when the level of confidence is increased.
What is decreased?
The change in the width of a confidence interval when the sample size is increased.
μ0
The claimed value of the population mean given in H0.
What is insufficient evidence to conclude alternative hypothesis is correct?
The conclusion you should make when P-value is greater than α.
What is believe or conclude alternative hypothesis is correct?
The conclusion you should make when P-value is less than α.
What is bias?
The condition eliminated by randomly allocating individuals to treatments.
What is replication?
The condition of having more than one individual in each treatment combination.
What is decreasing required sample size?
The effect of increasing desired margin of error on the required sample size.
Observed effect
The difference between the observed statistic and the claimed parameter value; e.g. x-bar = μ0.
Type II error
The error made when a false null hypothesis is not rejected. You fail to reject H0 when H0 is false.
Type I error
The error made when a true null hypothesis is rejected. You reject H0 when H0 is true.
What is blocking?
The grouping of experimental units according to some similar characteristic where the random allocation is carried out separately within each group.
What is null hypothesis?
The hypothesis of no change or no difference.
Null hypothesis
The hypothesis of no difference or no change. The hypothesis that the researcher assumes to be true until sample results indicate otherwise. Generally, the hypothesis that the researcher wants to disprove. (Note: Interpretations of P-value and statistically significant need to say something about "if H0 is true" in order to be correct.)
Matched pairs t test
The hypothesis testing method for matched pairs data. The typical null hypothesis is H0: μd = 0 where μd is the mean difference between treatments. For this test, a difference is computed within every pair. The mean and standard deviation of these differences are computed and used in computing the test statistic.
What is alternative hypothesis?
The hypothesis that the researcher wants to prove or verify.
Alternative hypothesis
The hypothesis that the researcher wants to prove or verify; a statement about the value of a parameter that is either "less than," "greater than," or "not equal to."
Margin of error for 95% confidence
The maximum amount that a statistic value will differ from the parameter value for the middle 95% of the distribution of all possible statistics. (Note: 95% can be changed to any other level of confidence.)
t*
The multiplier of standard error in computing margin of error for estimating a mean (or the difference between two means). The value for t* is found on the t table in the intersection of the appropriate df row and level of confidence column.
Level of confidence
The percent of the time that the confidence interval estimation procedure will give you intervals containing the value of the parameter being estimated. (Note: This can only be defined in terms of probability as follows: "The probability that the confidence interval to be computed (before data are gathered) will contain the value of the parameter." After data are collected, level of confidence is no longer a probability because a calculated confidence interval either contains the value of the parameter or it doesn't.)
β
The probability of failing to reject (accepting) a false null hypothesis.
P-value
The probability of getting a test statistic as extreme or more extreme than the value observed assuming H0 is true. OR The probability of obtaining a test statistic value as far or farther from the value actually obtained if H0 were true.
Power
The probability of rejecting a false null hypothesis; computed as 1 - β. Increase power by increasing sample size.
Level of significance (symbolized by α)
The probability of rejecting a true null hypothesis; equivalently, the largest risk a researcher is willing to take of rejecting a true null hypothesis.
Significance level (α)
The probability of rejecting a true null hypothesis; equivalently, the largest risk a researcher is willing to take of rejecting a true null hypothesis.
What is zero or one?
The probability that a computed confidence interval contains the value of the parameter it estimates.
What is zero or one? (Note: Think that this is a statement for P-value is a misconception.)
The probability that null hypothesis is true.
Zero or one
The probability that the value of the parameter μ is contained in the 95% confidence interval estimate for μ (e.g., the probability that the value of μ is contained in (26.7, 29.3) which is a 95% confidence interval estimate of μ is either zero or one.) OR the probability that the null hypothesis is true.
Variance
The square of standard deviation. Sample variance is s2 and population variance is σ2.
α
The symbol for level of significance.
Inference
Using results about sample statistics to draw conclusions about population parameters.
What is sampling distribution of x-bar?
What we use to find out the margin of error for estimating μ with a confidence interval.
What is margin of error?
With 90% confidence, the maximum amount that the statistic differs from the parameter for the middle 90% of all possible statistics.
Estimated standard deviation of x-bar
called standard error of x-bar and equals s divided by square root n; measures variability of sampling distribution of x-bar.
Symbols for statistics
x-bar, s
Symbols for parameters
μ, σ
What is the parameter for comparing two population means?
μ1 - μ2
