Examples of z-test in the following topics:
-
- 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 $Z = \frac{(T-\theta)}{s},fromwhichone−tailedandtwo−tailedp−valuescanbecalculatedas\varphi(-Z)$ (for upper-tailed tests), $\varphi(Z)$ (for lower-tailed tests) and $2\varphi(\left|-Z\right|)$ (for two-tailed tests) where φ 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.
-
- 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.
-
- 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 $z−statistic,t−statistic,chi−squarestatistic,andF$-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 $z−statisticforuseinaone−samplez$-test is as follows:
-
- 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 $t=\frac{Z}{s},whereZands$ 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).
-
- 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).
-
- 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.
-
- 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.
-
- 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 Nr≥10, a $z$-score can be calculated as follows:
-
- 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.
-
- Thus, a positive $z−scorerepresentsanobservationabovethemean,whileanegativez$-score represents an observation below the mean.
- $z−scoresarealsocalledstandardscores,z$-values, normal scores or standardized variables.
- The use of "$z"isbecausethenormaldistributionisalsoknownasthe"z$ 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