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Hypothesis Testing (Normal Distribution) (Cambridge (CIE) A Level Maths: Probability & Statistics 2): Flashcards

Exam code: 9709

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FrontHypothesis Testing for the Mean of a Normal Distribution
In a test on a population mean, which distribution does the test actually work with?

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In a test on a population mean, which distribution does the test actually work with?
The distribution of the sample mean $\bar{X}$ , not the population distribution $X$ the question quotes.
Assuming $H_{0}$ is true, $\bar{X} ~ N (μ, \frac{σ^{2}}{n})$ , and the observed sample mean is judged against that.
Using the population distribution instead treats a mean of $n$ readings as though it were a single reading.
Complete the test statistic for a hypothesis test on the mean of a normal distribution.
$z = \frac{\bar{x} - μ}{\frac{σ}{______}}$
The completed test statistic is:
$z = \frac{\bar{x} - μ}{\frac{σ}{\sqrt{n}}}$
The denominator is the standard deviation of $\bar{X}$ , so it carries $\sqrt{n}$ rather than $n$ .
For $X ~ N (204, 81)$ and a sample of 12, what is the denominator of $z$ ?
It is $\frac{9}{\sqrt{12}} = 2.598$ to three decimal places.
The 81 is the variance, so the standard deviation is $\sqrt{81} = 9$ , and it is the standard deviation that goes into the formula.
Putting 81 in the numerator of the fraction is the commonest slip here.
How is the critical $z$ value found for a 5% one-tailed test?
Read it from the table of critical values for the standard normal distribution, which gives $z = 1.645$ at 5%.
Use $- 1.645$ when the test is for a decrease, since the critical region is then in the lower tail.
True or False?
A test statistic of $z = - 1.15$ does not lie in the critical region of a 5% lower-tailed test.
True.
The critical value is $- 1.645$ , and $- 1.15$ is closer to zero than that, so it sits outside the region.
Compare how far each value is from 0 rather than which is numerically larger, or the negative signs will reverse the decision.
How is the critical value found as a value of $\bar{x}$ ?
Set the test statistic equal to the critical $z$ from the table, then solve for $\bar{x}$ .
For $N (204, 81)$ with $n = 12$ at 5% for a decrease, $- 1.645 = \frac{\bar{x} - 204}{2.598}$ gives $\bar{x} = 199.7$ .
The observed sample mean can then be compared directly, in the units of the question.
What are the two critical values in a two-tailed test on a normal mean?
The positive and negative $z$ for half the significance level, which have the same size because the normal distribution is symmetrical.
Solving with each in turn gives two values of $\bar{x}$ , and they should come out the same distance either side of the mean.
That symmetry is a useful check on the arithmetic.

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