Hypothesis Testing (Z Test)


This test can be applied if the sample size is large, or if the sample is drawn from a normal distribution of known variance.

If the population variance is unknown, it can be estimated using:
σ² = (1 / (n-1) ) ( Σx² - (Σx)² / n )

Exercises for students
Solve the following problems on paper and check your working using the above applet:

Given sample size = 30, sample mean = 52, population variance = 30, test, at the 5% level, the hypothesis that the population mean = 50 against the alternative hypothesis that the pop. mean > 50.
Repeat at the 10%, 2% and 1% levels.
Repeat using sample mean = 51 & 53.
Test against the alternative hypothesis that pop. mean is not equal to 50.

Given sample size = 30, sample mean = 48, population variance = 30, test, at the 5% level, the hypothesis that the population mean = 50 against the alternative hypothesis that the pop. mean < 50.
Repeat at the 10%, 2% and 1% levels.
Repeat using sample mean = 47 & 49.
Test against the alternative hypothesis that pop. mean is not equal to 50.

Given sample size = 30, sample proportion = 0.7, population variance = 30, test, at the 5% level, the hypothesis that the population proportion = 0.5 against the alternative hypothesis that the pop. proportion < 0.5.
Repeat at the 10%, 2% and 1% levels.
Repeat using sample proportion = 0.6 & 0.8.
Test against the alternative hypothesis that pop. proportion is not equal to 0.5.


The length of wool in the balls of wool produced by a certain manufacturer has mean µ m and standard deviation 6 m. The manufacturer claims that µ = 50.
A random sample of 30 balls of wool is taken and the sample mean is found to be 48 m.
Test whether this provides significant evidence, at the 5% level, that the manufacturer has overstated the value of µ.


The yield of potatoes per hectare of land is normally distributed with mean 10.0 tonnes and standard deviation 2 tonnes.
A new farming method is introduced. 30 randomly chosen plots of land is tested and the mean yield per hectare is found to be 10.7 tonnes.
Is there any evidence of significant improvement? Test at the 2% level.



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