### Overview of Statistical Hypotheses

A statistical hypothesis is a claim about a population parameter. For example,

- The mean height of males is less than 65 inches tall
- The percentage of people favoring a bullet train is about 59%
- The daily average expense for a college student is more than Rs. 250
- At least 5% of Pakistan earn more than Rs 2,500,000 per year

A statistical method is used to determine if there is enough evidence in sample data to support a claim about a population.

The claimed hypotheses are written in certain statistical and concise forms. For example, the above statements about population can be written as

- $H_0: \mu < 65$
- $H_0: \pi = 0.59$
- $H_0:\mu > 250$
- $H_0: \pi \ge 0.05$

If someone is interested in knowing that above stated statistical hypotheses are either true or false, one needs to conduct a hypothesis test. To test a statistical hypothesis, one needs to follow the following basic procedure, to fulfill the requirements.

- Draw a random sample from the population of interest (for example, the height of males)
- Determine if the results from the sample data are consistent or not with the hypothesis under study.
- If the collected sample data is (significantly) different from the claimed hypothesis, then reject the hypothesis as being false. However, if the sampled data is not significantly different, one would not reject the hypothesis.

### Statistical Hypotheses Example

**Example:** Suppose a battery manufacturer claims that the average life of their batteries is at least 300 minutes.

To test this hypothesis, we follow the procedure as

- Select a sample of say $n=100$ batteries. The sample of batteries is tested and the mean life of sampled batteries was found to be $\overline{x} = 294$ minutes with a sample standard deviation of $s=204 minutes.
- We need to test whether “is this data sufficiently different from the manufacturer’s claim to justify rejecting the claim as false”?
- Since the sample drawn is large enough, the Central Limit Theorem allows us to conclude that the distribution of sample means $\overline{x}$ is approximately normal.
- If the manufacturer’s claim is correct, then $\mu_{\overline{x}} = \mu \ge 300$ and we will assume that $\mu_{\overline{x}} = \mu = 300$.
- The Z-score will be $$Z = \frac{\overline{x} – \mu_{\overline{x}}}{\frac{s}{\sqrt{n}}}=\frac{294-300}{\frac{20}{\sqrt{100}}} = -3.0$$
- Search the Probability value from Standard Normal Table, as $P(\overline{x} \le 294)=0.0013$

### Decision about Hypothesis

Now one of the following must be true

- The assumption that $\mu = 300$ is incorrect
- The sample drawn has a so small mean that only 13 in 10,000 samples have a mean as low.

The probability of the second statement being true is quite small (0.0013). Thus there is strong evidence to believe that the first statement is true, and hence the manufacturer overstated the mean life of their batteries.