Edexcel IAL - Statistics 3- 3.5 Hypothesis Tests for the Mean (Variance Known)- Study notes  - New syllabus

Hypothesis Tests for the Mean of a Normal Distribution (Variance Known)

Hypothesis testing is a formal procedure used to decide whether there is sufficient evidence from sample data to support a claim about a population parameter. In this section, we consider tests for the population mean when the population variance is known and the population is normally distributed.

Assumptions

The following conditions apply:

• The population distribution is normal
• The population variance \( \mathrm{\sigma^2} \) is known
• A random sample of size \( \mathrm{n} \) is taken

Null and Alternative Hypotheses

A hypothesis test begins with two competing hypotheses:

Null hypothesis \( \mathrm{H_0} \): the population mean has a specified value

Alternative hypothesis \( \mathrm{H_1} \): the population mean differs from the value in \( \mathrm{H_0} \)

Common forms of the alternative hypothesis are:

Two-tailed: \( \mathrm{H_1:\mu \neq \mu_0} \)

Upper-tailed: \( \mathrm{H_1:\mu > \mu_0} \)

Lower-tailed: \( \mathrm{H_1:\mu < \mu_0} \)

Test Statistic

The test statistic used is

\( \mathrm{Z = \dfrac{\bar{X} – \mu_0}{\sigma/\sqrt{n}}} \)

Under the null hypothesis, this statistic has the distribution

\( \mathrm{Z \sim N(0,1)} \)

Significance Level

The significance level, denoted by \( \mathrm{\alpha} \), is the probability of rejecting the null hypothesis when it is true.

Common significance levels are:

5% (\( \mathrm{\alpha = 0.05} \))

1% (\( \mathrm{\alpha = 0.01} \))

Critical Region Method

Using the critical region approach:

For a two-tailed test at 5% level: reject \( \mathrm{H_0} \) if \( \mathrm{|Z| > 1.96} \)

For an upper-tailed test at 5% level: reject \( \mathrm{H_0} \) if \( \mathrm{Z > 1.645} \)

For a lower-tailed test at 5% level: reject \( \mathrm{H_0} \) if \( \mathrm{Z < -1.645} \)

p-value Method

Alternatively, the p-value may be used.

The p-value is the probability of obtaining a test statistic at least as extreme as the observed value, assuming \( \mathrm{H_0} \) is true

Reject \( \mathrm{H_0} \) if the p-value is less than \( \mathrm{\alpha} \)

Link with Confidence Intervals

For a two-tailed test at significance level \( \mathrm{\alpha} \):

Reject \( \mathrm{H_0} \) if the hypothesised mean \( \mathrm{\mu_0} \) lies outside the \( \mathrm{(1-\alpha)\times100\%} \) confidence interval

Do not reject \( \mathrm{H_0} \) if it lies inside the interval

Key Points to Remember

• The test statistic follows \( \mathrm{N(0,1)} \) under \( \mathrm{H_0} \)
• Known variance allows direct use of the normal distribution
• Decisions are based on critical values or p-values