Testing the mean of a sample by the t-test and using a normal distribution
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Testing the mean of a sample drawn from a normal distribution using a t&-test and using a no
[Equation goes here &- download the original to see it.] In that case we can use the usual test statistic [Equation goes here &- download the original to see it.] The critical value is found in the usual way from the standardised z&-tables. Example (1) The volumes of cans produced by a certain machine are normally distributed with mean volume 502cm3 and standard deviation 14cm3. Following a service a sample of 80 cans is taken and it is found that the sample mean is now 497cm3. Is there any evidence at the 5% level that the service has caused a change in the mean volume of the cans manufactured by the machine? [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] z test > z critical Reject H0, accept H1 The service has altered the mean values of the cans. The above example deals with the case where the variance is known. Furthermore, it applies where either the population is known to be normally distributed or the sample size is [Equation goes here &- download the original to see it.]. However, what do we do when the variance is not known and the sample size is small[Equation goes here &- download the original to see it.] The simple answer is that we have to conduct the same test, but substitute the unbiased estimate of the population variance derived from the sample itself. However, a complication arises. There is a possibility that the unbiased sample variance may not be accurate. Hence, we need to use a probability distribution that accounts for this. Intuitively, it would be a modification of the standardised normal distribution that makes adjustments for the increased possibility that the "tails" of the distribution would be fatter because of the additional source of random error. The distribution that does this is called the "Student&'s t". It was discovered by William Goset a statistician working for Guiness. Guiness did not permit publication of work connected to the company, so Goset decided to publish under the pseudonym of Student. That is why it is known as "Student&'s t". A comparison of the t&-distribution for low sample values with the z&- distribution would give something like this [Diagram goes here &- download the original to see it.] With this adjustment, the t&-test is like the z&-test, and the test is conducted in exactly the same way, with the tables for Student&'s&-t used in place of those for the standardised normal variable, see next page.
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Contents of
Testing the mean of a sample by the t-test and using a normal distribution
1 Testing the mean of a sample drawn from a normal distribution using a t&-test and using a no
2 Summary of testing the mean of a sample drawn from a normal distribution using a t&-test and
3 Example of testing the mean of a sample drawn from a normal distribution using a t&-test and
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