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Z-tests are used to compare the collected data of a defined population and the sample. The z-score tells you how far, in standard deviation, a data point is from the mean or average of a data set. A z-test compares a sample to a defined population. Normally, a z-test identifies issues with larger samples that are being studied (n > 30). Z-tests are also useful when testing out a hypothesis. Generally, they are most useful when the standard deviation is known.
T-tests are calculations used to test a hypothesis. However, t-tests are used to determine the statistical difference between two independent sample groups. In other words, a t-test analyzes how likely the difference between two samples occurs due to random chance. Usually, t-tests are most appropriate to use when dealing with problems with a limited or small sample size (n < 30).
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One-sample T-tests are used to compare the mean average between the sample and the population studied. The test is also applied compare the difference between the mean of the sample and the mean of the population. Z-tests always use normal distribution and should be used when the standard deviation is known after the data is collected.
A z-score and a t- score are both used in hypothesis testing. Generally, in elementary stats and AP stats, you’ll use a z-score in testing more often than a t score.
Z-scores are a conversion of individual scores into a standard form. The conversion is based on your knowledge about the population’s standard deviation and mean. A z-score tells you how many standard deviations from the mean your result is. You can use your knowledge of normal distributions (like the 68 95 and 99.7 rule) or the z-table to determine what percentage of the population will fall below or above your result. Like z-scores, t-scores are also a conversion of individual scores into a standard form. However, t-scores are used when you don’t know the population standard deviation.
You use the z-score test if:
Population normal and variance known (for any sample size)
1. Population normal, variance unknown and n > 30 n>30 (due to CLT)
2. Population binomial, n p > 10 np>10, n q >10 nq>10
You use the t- score test if:
1. Population normal, variance unknown and n < 30 n<30
2. No knowledge about population or variance and n < 30 n<30, but sample data looks normal / passes tests etc. so population can be assumed normal.