Statistics T-test/ANOVA

A tire manufacturer wishes to determine whether, on average, a steel belted radial tire provides more than 50,000 miles of wear. A random sample yields a sample mean of 53,500 miles and a standard deviation of 5,300 miles. From the manufacturer's perspective, it would be best if the sample size were

50

Given the following ANOVA summary table...

55

In an experiment involving three different groups, each consisting of 10 subjects, the degrees of freedom for within groups equals

27

In an experiment involving four different groups, each consisting of 5 subjects, the degrees of freedom for between groups equals

3

In an experiment involving four different groups, each consisting of 8 subjects, the critical F occupies the cell intersected by column and row degrees of freedom, respectively, of

3 and 28

Given that three gas mileage tests yield a mean of 30 miles per gallon, and that two of the tests have values of 27 and 29, the third test must have a value of

34

Given absolute differences of 5, 3, 6, 1, 5, and 8 for six pairs of means, and a critical HSD value of 4.23, it would be correct to conclude that the number of significant differences equals

four

Even though the population standard deviation is unknown, an investigator uses z rather than the more appropriate t to test a hypothesis at the .01 level of significance. In this situation the true level of significance is

larger than .01.

To save space, t tables supply only critical values of t for a few of the more common

levels of significance

In analysis of variance, variance estimates often are referred to as

mean squares.

Use analysis of variance rather than a t test whenever the null hypothesis makes a claim about

more than two population means

Before the null hypothesis can be rejected at a given level of significance, observed values of the t ratio (compared to those for the z ratio)

must deviate further from zero

Even though there are n deviations in a sample, only n 1 of these deviations supply a true picture of the population deviations because only n 1 are free to vary, given that the sum of

n deviations from their own sample mean always equals zero.

In the analysis of variance, the F test is equivalent to a

nondirectional test because all variations are squared.

Essentially, the F ratio reflects the ratio of the

observed sample mean differences to the estimated error term.

On the basis of a random sample of 25 students from a large university, an educator estimates, with 95 percent confidence, that between 17 to 20 hours describes the mean weekly study time of all students at the university. Accordingly, we can be reasonably certain that

on average, all students study between 17 and 20 hours

Degrees of freedom refer to the number of values that are free to vary, given

one or more mathematical restrictions

Regardless of whether the null hypothesis is true or false, variability within groups reflects

only random error.

The estimated standard error of the mean is to be used whenever you must estimate the

population standard deviation.

The tables for t show

positive critical t values for a family of t distributions.

A property of all hypothesis tests is that

rare outcomes cause the null hypothesis to be rejected.

If the observed F equals or exceeds the critical F, the experimental outcome is

rare, and the null hypothesis is rejected

If a published report of an F test specified that p <.01, you could conclude that the test result is

rare, supporting the research hypothesis.

Given a critical t of and an observed t of , you should

retain the null hypothesis.

Even when the normality assumption is violated, the t test retains much of its accuracy as long as the

sample size isn't too small.

When using the t test for a single population mean, a single degree of freedom is lost. Therefore one is subtracted from the

sample size.

You needn't be too concerned about violating the assumptions of the F test so long as

sample sizes are equal and larger than 10.

Compared to the normal distribution, a distinctive property of the t distribution is its

Inflated Tails

The t distribution is most different from the standard normal distribution when sample size is

Large

Cohen's d requires that a mean difference be divided by the square root of the

MSerror

In a table that summarizes the results of an analysis of variance, the word "Between" might be replaced by

Treatment

An F test of the null hypothesis is based on the notion that if the null hypothesis is true, the numerator of the F ratio tends to be

about the same as its denominator

Rather than being concerned about the single multiple comparison test that is most appropriate in a particular application, you could initiate with the aid of a computer program

an entire series of multiple comparison tests

Use Cohen's d to estimate the effect size of

any significant difference between pairs of means

A treatment effect exists if differences exist between

at least one pair of population means.

Rejection of the overall null hypothesis indicates that

at least one population mean differs from all others.

In analysis of variance, observed mean differences appear, somewhat disguised, as variability

between groups

In those cases where the status of a particular comparison is ambiguous, being designated as significant by some multiple comparison tests and as nonsignificant by the remaining tests, it is suggested that this comparison be reported as having

borderline significance.

Rejection of the overall null hypothesis usually raises some additional questions regarding

both a and b

If the null hypothesis is false, variability between groups reflects

both random error and the treatment effect

Tukey's HSD controls the

cumulative probability of a type I error

If the null hypothesis is false because of a sizable treatment effect, the value of F tends to be

considerably larger than one.

The major difference between a z ratio and a t ratio is that the t ratio

contains extra variability in its denominator.

If the value of the observed F is less than one -- say, it equals 0.83 -- you can conclude that the null hypothesis

could be true

Given a significant F test, the size of the overall effect can be estimated by using a measure equal to the squared

curvilinear correlation between independent and dependent variables.

If the sum of squares for between groups equals 50 and that for within groups equals 70, the sum of squares for total variability must

equal 120.

Each t distribution is associated with a special number that directly reflects the

degrees of freedom

The squared curvilinear correlation indicates the proportion of variance in

dependent variable attributable to the independent variable

It is recommended that reports in the literature describe not only the results of statistical tests and estimates of effect size, but also

descriptions of group means and standard deviations.

To obtain the estimated standard error, divide the sample standard deviation by

square root of n

In the Minitab output for one-factor ANOVA, each of the three sample means generates a 95 percent confidence interval based on the pooled standard deviation or

square root of the within groups or error mean square

Variability between groups is based on the variation among the scores of

subjects treated differently.

In analysis of variance, variance estimates consist of

sum of squares divided by degrees of freedom.

Any sum of squares term always equals the

sum of the squared deviations of all scores about their mean

The loss of a single degree of freedom occurs for the t test for a single population mean because only n 1 of the observations

supply information about population variability

If the desired number of degrees of freedom doesn't appear in the t table, use the critical t associated with

the next smallest df in the table.

Tukey's test should be used only if

the overall null hypothesis is rejected

You might choose the most appropriate multiple comparison test (from among the many possible tests) depending on

the relative seriousness of type I and II errors.

Cohen's guidelines for values of d associated with small, medium, and large effects are

the same as for both two independent or two related samples.

To find the appropriate t value for a 95 percent confidence interval, read the t table entry under the column heading that also describes a hypothesis test for a

two tailed test, .05 level of significance.

When all possible differences between pairs of population means are evaluated not with an F test, but with a series of regular t tests, the probability of at least one

type I error is larger than the specified level of significance

To pinpoint specific differences between pairs of population means

use a multiple comparison test.

If a treatment effect exists,

variability between groups will tend to exceed variability within groups

The squared curvilinear correlation is obtained by dividing the

within sum of squares by the between sum of squares.

When there is very extensive overlap between scores for all groups, the value of the squared curvilinear correlation tends toward

zero