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# Statistics

## Identifying the Test Statistic

There are many different test statistics to choose from. It depends on what parameter you are testing (e.g., $$\mu,\sigma,p$$), what variables are given (is $$\sigma$$ known?), and the distribution of the population (e.g., normally distributed). The following are some test statistics you will encounter for hypothesis testing with one-sample.

 Parameter Sampling Distribution Requirements Test Statistic Proportion p Normal (z) $$np\geq 5$$ and $$nq \geq 5$$ $z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}$ Mean $$\mu$$ t $$\sigma$$ is not known and normally distributed population or $$\sigma$$ not known and $$n\leq 30$$ $t=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}$ Mean $$\mu$$ Normal (z) $$\sigma$$ is known and normally distributed population or $$\sigma$$ known or $$n>30$$ $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$ Standard deviation $$\sigma$$ $$\chi^2$$ Strict requirement: normally distributed population $\chi^2=\frac{(n-1)s^2}{\sigma^2}$

Example 1: 93 student course evaluations report an average rating of 3.91 with standard deviation 0.53. What test statistic should be used for to test the hypothesis that the population student course evaluations has a mean equal to 4.00?

Solution: The given values are $$\bar{x}=3.91$$, $$s=0.53$$, and $$\mu=4.00$$. $$\sigma$$ is not known and $$n>30$$ so the t-statistic should be used.

\begin{align} t&=\frac{\bar{x}-\mu}{\frac{s}{\sqrt{n}}}\\ &=\frac{3.91-4.00}{\frac{0.53}{\sqrt{93}}}\\&=-1.63760\end{align}

Example 2: A study of of 19,136 people found that 29.2% of the people sleep-walked. Would a reporter be justified in stating that "fewer than 30% of adults have sleep-walked"? What test statistic should be used?

Solution: The given value is in proportions with $$\hat{p}=0.292$$, $$p=0.30$$, which means $$q=1-p=0.7$$.

First we have to meet the requirements $$np\geq 5$$ and $$nq \geq 5$$ to apply the test statistic.

$$np=(19,136)(0.3)=5740.8\geq 5$$ and $$nq=(19,136)(0.7)=13,395.2\geq 5$$

With the conditions satisfied we can calculate the test statistic for proportions.

\begin{align} z&=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}\\ &=\frac{0.292-0.30}{\sqrt{\frac{(0.3)(0.7)}{19,136}}} \\ &=-2.41494\end{align}

## Examples

In a test of the effectiveness of garlic for lowering cholesterol, 49 subjects were treated with raw garlic. Cholesterol levels were measured before and after the treatment. The changes (before minus after) in their levels of LDL cholesterol (in mg/dL) have a mean of 0.4 and a standard deviation of 21.0. Use a 0.05 significance level to test the claim that with garlic treatment, the mean change in LDL cholesterol is greater than 0. What do the results suggest about the effectiveness of garlic treatment?

See the video below to follow each step of hypothesis testing to solve the problem above using the P-value method.

Crash tests were conducted for child booster seats for cars. Listed below are results from one of those tests, with the measurements given in hic (standard head injury condition units). The safety requirement is that the hic measurement should be less than 1000 hic. Use a 0.01 significance level to test the claim that the sample is from a population with a mean less than 1000 hic. Do the results suggest that all of the child booster seats meet the specified requirement?

774     649     1210      546     431     612

See the video below to follow each step of hypothesis testing to solve the problem above using the critical value method.