Z-Tests - Maple Help

Z-Tests

 A company produces metal discs with a mean weight of grams and standard deviation ofgrams.   Suppose that the company takes a sample of size 50 and finds that the sample mean is.   Assuming a significance level of 95%, is the company correct in accepting the null hypothesis that the sample does not have different weights on average than the population of metal discs?

Background

The Z-test is used to compare means of two distributions with known variance. One sample Z-tests are useful when a sample is being compared to a population, such as testing the hypothesis that the distribution of the test statistic follows a normal distribution. Two-sample Z-tests are more appropriate for comparing the means of two samples of data.

Requirements for the Z-test:

 • The mean and standard deviation of the population distribution are known.
 • The mean of the sample distribution is known.
 • The variance of the sample is assumed to be the same as the population.
 • The population is assumed to be normally distributed
 • The population size is over 30

In cases where the population variance is unknown, or the sample size is less than 30, the Student's t-test  may be more appropriate.

To calculate a Z-test statistic, the following formula can be used:

z = $\frac{x\mathit{-}\mathrm{μ}}{\mathrm{SE}}$,

z = $\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$,

where x is the sample mean, m is the population mean, and SE is the standard error, which can be calculated using the following formula:

SE = $\frac{\mathrm{σ}}{\sqrt{n}}$,

where s is the population standard deviation and n is the sample size.

For each significance level, α, the Z-test has a critical value. For example, the significance level α = 0.01, has a critical value of 2.326. If the Z-test statistic is greater than this critical value, this may provide evidence for rejecting the null hypothesis.



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