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How to Do Hypothesis Testing with Normal Distribution

Hypothesis tests compare a result against something you already believe is true. Let $X_{1}, X_{2}, \dots, X_{n}$ be $n$ independent random variables with equal expected value $μ$ and standard deviation $σ$ . Let $\overset{}{X}$ be the mean of these $n$ random variables, so

\overset{}{X} = \frac{1}{n} \sum_{i = 1}^{n} X_{i}

The stochastic variable $\overset{}{X}$ has an expected value $μ$ and a standard deviation $\frac{σ}{\sqrt{n}}$ . You want to perform a hypothesis test on this expected value. You have a null hypothesis $H_{0} : μ = μ_{0}$ and three possible alternative hypotheses: $H_{a} : μ < μ_{0}$ , $H_{a} : μ > μ_{0}$ or $H_{a} : μ \neq μ_{0}$ . The first two alternative hypotheses belong to what you call a one-sided test, while the latter is two-sided.

In hypothesis testing, you calculate using the alternative hypothesis in order to say something about the null hypothesis.

Rule

Hypothesis Testing (Normal Distribution)

1.: You set up the null hypothesis $H_{0}$ against an alternative hypothesis $H_{A}$ . $H_{0} : μ = μ_{0}$ against $H_{a} : μ > μ_{0}$ (possibly $H_{a} : μ < μ_{0}$ , $H_{a} : μ \neq μ_{0}$ ).
2.: Then, you do an experiment and find that the average value is $\overset{}{x}$ . Next, you calculate the probability $P (\overset{}{X} \geq \overset{}{x})$ for the alternative hypothesis $H_{a} : μ > μ_{0}$ .
3.: If this probability is less than $1$ %, $5$ % or $10$ %, you discard $H_{0}$ .

Note! For two-sided testing, multiply the

p

-value by 2 before checking against the critical region.

Example 1

As the production manager at the new soft drink factory, you are worried that the machines don’t fill the bottles to their proper capacity. Each bottle should be filled with $0.5 L$ soda, but random samples show that 48 soda bottles have an average of $0.48 L$ , with an empirical standard deviation of $0.1$ . You are wondering if you need to recalibrate the machines so that they become more precise.

This is a classic case of hypothesis testing by normal distribution. You now follow the instructions above and select $10$ % level of significance, since it is only a quantity of soda and not a case of life and death.

1.

The null hypothesis is that there is

0.5

L in each bottle:

μ_{0} = 0.5

The alternative hypothesis in this case is that the bottles do not contain $0.5$ L and that the machines are not precise enough. This thus becomes a two-sided hypothesis test and you must therefore remember to multiply the $p$ -value by 2 before deciding whether the $p$ -value is in the critical region. This is because the normal distribution is symmetric, so $P (X \geq k) = P (X \leq - k)$ . Thus it is just as likely to observe an equally extremely high value as an equally extreme low:

μ_{0} \neq 0.5

2.

Find the

p

-value by calculating

P (\overset{}{X} \geq 0.48)

\begin{array}{l} P (\overset{}{X} \leq 0.48) & = P (Z \leq \frac{0.48 - 0.5}{\frac{0.1}{\sqrt{48}}}) \\ = P (Z \leq - 1.39) \\ = 0.0823 \\ ⇕ \\ p & = 0.0823 \cdot 2 \\ = 0.1646 \\ = 16.46 % \end{array}

3.

You have decided that if the machines are to be recalibrated your

p

-value must be less than

10

%. Therefore, there must be less than

10

% chance of the machine filling

0.5

L on average in the bottles. Your

p

-value is

p = 16.46 % > 10 %,

so $H_{0}$ must be kept, and the machines are deemed to be fine as is.

Had the $p$ -value been less than the level of significance, that would have meant that the calibration represented by the alternative hypothesis would be significantly better for the business.

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