z-test

(noun)

Any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution.

Related Terms

  • t-test

Examples of z-test in the following topics:

  • When Does the Z-Test Apply?

    • A $z$-test is a test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution.
    • We then calculate the standard score $Z = \frac{(T-\theta)}{s}$, from which one-tailed and two-tailed $p$-values can be calculated as $\varphi(-Z)$ (for upper-tailed tests), $\varphi(Z)$ (for lower-tailed tests) and $2\varphi(\left|-Z\right|)$ (for two-tailed tests) where $\varphi$ is the standard normal cumulative distribution function.
    • For larger sample sizes, the $t$-test procedure gives almost identical $p$-values as the $Z$-test procedure.
    • For the $Z$-test to be applicable, certain conditions must be met:
    • If the variation of the test statistic is strongly non-normal, a $Z$-test should not be used.
  • Quantitative or Qualitative Data?

    • Different statistical tests are used to test quantitative and qualitative data.
    • Paired and unpaired t-tests and z-tests are just some of the statistical tests that can be used to test quantitative data.
    • A z-test is any statistical test for which the distribution of the test statistic under the null hypothesis can be approximated by a normal distribution.
    • For each significance level, the z-test has a single critical value.
    • Therefore, many statistical tests can be conveniently performed as approximate z-tests if the sample size is large or the population variance known.
  • One, Two, or More Groups?

    • A test statistic is considered to be a numerical summary of a data-set that reduces the data to one value that can be used to perform a hypothesis test.
    • Examples of test statistics include the $z$-statistic, $t$-statistic, chi-square statistic, and $F$-statistic.
    • A $z$-statistic may be used for comparing one or two samples or proportions.
    • When comparing two proportions, it is necessary to use a pooled standard deviation for the $z$-test.
    • The formula to calculate a $z$-statistic for use in a one-sample $z$-test is as follows:
  • Assumptions

    • Assumptions of a $t$-test depend on the population being studied and on how the data are sampled.
    • Most $t$-test statistics have the form $t=\frac{Z}{s}$, where $Z$ and $s$ are functions of the data.
    • Typically, $Z$ is designed to be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas $s$ is a scaling parameter that allows the distribution of $t$ to be determined.
    • This can be tested using a normality test, or it can be assessed graphically using a normal quantile plot.
    • If using Student's original definition of the $t$-test, the two populations being compared should have the same variance (testable using the $F$-test or assessable graphically using a Q-Q plot).
  • Exercises

    • In general, are rank randomization tests or randomization tests more powerful?
    • What is the advantage of rank randomization tests over randomization tests?
    • (S) Test the difference in central tendency between the two conditions using a rank-randomization test (with the normal approximation) with a one-tailed test.
    • Give the Z and the p.
    • (SL) Test the difference in central tendency between the four conditions using a rank-randomization test (with the normal approximation).
  • The chi-square test statistic

    • In previous hypothesis tests, we constructed a test statistic of the following form:
    • These two ideas will help in the construction of an appropriate test statistic for count data.
    • That is, Z 1, Z 2, Z 3, and Z 4 must be combined somehow to help determine if they – as a group – tend to be unusually far from zero.
    • |Z 1 | + |Z 2 | + |Z 3 | + |Z 4 | = 4.58
    • The test statistic X 2 , which is the sum of the Z 2 values, is generally used for these reasons.
  • Normal probability table

    • She would like to know what percentile she falls in among all SAT test-takers.
    • A normal probability table, which lists Z scores and corresponding percentiles, can be used to identify a percentile based on the Z score (and vice versa).
    • We can also find the Z score associated with a percentile.
    • We determine the Z score for the 80th percentile by combining the row and column Z values: 0.84.
    • Determine the proportion of SAT test takers who scored better than Ann on the SAT.
  • Wilcoxon t-Test

    • It can be used as an alternative to the paired Student's $t$-test, $t$-test for matched pairs, or the $t$-test for dependent samples when the population cannot be assumed to be normally distributed.
    • The test is named for Frank Wilcoxon who (in a single paper) proposed both the rank $t$-test and the rank-sum test for two independent samples.
    • In consequence, the test is sometimes referred to as the Wilcoxon $T$-test, and the test statistic is reported as a value of $T$.
    • Other names may include the "$t$-test for matched pairs" or the "$t$-test for dependent samples."
    • Thus, for $N_r \geq 10$, a $z$-score can be calculated as follows:
  • 68-95-99.7 rule

    • SAT scores closely follow the normal model with mean µ = 1500 and standard deviation σ = 300. ( a) About what percent of test takers score 900 to 2100?
    • To find the area between Z = −1 and Z = 1, use the normal probability table to determine the areas below Z = −1 and above Z = 1.
    • Repeat this for Z = −2 to Z = 2 and also for Z = −3 to Z = 3.
    • 3.23: (a) 900 and 2100 represent two standard deviations above and below the mean, which means about 95% of test takers will score between 900 and 2100.
    • (b) Since the normal model is symmetric, then half of the test takers from part (a) (95% / 2 = 47.5% of all test takers) will score 900 to 1500 while 47.5% score between 1500 and 2100.
  • Z-Scores and Location in a Distribution

    • Thus, a positive $z$-score represents an observation above the mean, while a negative $z$-score represents an observation below the mean.
    • $z$-scores are also called standard scores, $z$-values, normal scores or standardized variables.
    • The use of "$z$" is because the normal distribution is also known as the "$z$ distribution."
    • This may include, for example, the original result obtained by a student on a test (i.e., the number of correctly answered items) as opposed to that score after transformation to a standard score or percentile rank.
    • Define $z$-scores and demonstrate how they are converted from raw scores
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