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Hypothesis Testing

In the previous section, the emphasis was on the determination of various population parameters. This section deals with testing claims (or hypothesis) made about population parameters.

Definition

In statistics a hypothesis is a claim or statement about a property of a population.

An Example of validating a hypothesis

Coin flipper Inc., has introduced a machine, the Coin Flipper, that will flip a coin. There claim is that "the machine will flip the coin to land on heads up to 85% of the time." What would you conclude if out of 100 tosses:

1. 55 heads?
2. 96 heads?

Solution:

1. With out the machine tossing the coin, it is expected that there would be 50 heads out of 100 tosses. The result of 55 heads, while not the expected value, is not that unreasonable. The fact that 55 heads were produced could have happened by chance, does not validate the companies claim.
2. The chances of getting 96 heads out of 100 tosses seems VERY unlikely. In this case, the likely explanation is that the companies Coin Flipper is effective.

Formal Hypothesis Test

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When conducting a hypothesis test, there are standard components that need to be defined.

• The null hypothesis (denoted $H_0$) is a statement about the population statistic, such as $\mu$.

• The alternate hypothesis (denoted $H_1$) is a claim to be tested.

  There are three ways to set up a the null and alternate hypothesis.

  1. Equal hypothesis verses not equal hypothesis (two-tailed test)
     $H_0$:$\mu$ = some value
     $H_1$:$\mu$ ≠some value

  2. Equal hypothesis verses less than hypothesis (left-tailed test)
     $H_0$:$\mu$ = some value
     $H_1$:$\mu <$ some value

  3. Equal hypothesis verses greater than hypothesis (right-tailed test)
     $H_0$:$\mu$ = some value
     $H_1$:$\mu >$ some value

• A significance level (denoted by $\alpha$). This is the same $\alpha$ as in the degree of confidence from the section on confidence intervals.

• The critical region is the region under the normal curve that cause a rejection of the null hypothesis. This region can be left-tailed, right-tailed or two-tailed.

• The test statistic is a value that is computes from the sample data that is used to in making the decision about the rejection of the null hypothesis. The test statistic is found by the formula

  $$z=\dfrac{\bar{x} - \mu}{\dfrac{\sigma}{\sqrt{n}}}$$ (6.1)

  
  

If you want to use a hypothesis to support your claim, then the claim MUST be worded so that it beomes the alternate hypothesis.

Example

Suppose 150 college students were asked their Intelligence Quotient (IQ). From the data you find out that x=120 and $\sigma$=11.3. A college professor says that he knows the population mean, $\mu$ is less than 116. Using a 0.05 significance level, determine if the professor may be correct. Solution: The people running the project want the professor to be wrong, so you want to prove the claim "The population mean is greater than 116." (Remember that only the alternate hypothesis can be supported, so the statement should NOT contain an equality.) So in this case let:

$H_0$:	$\mu$ = 116
$H_1$:	$\mu$ > 116

With the null hypothesis $H_0$: $\mu$ = 116, the critical region is in the right tail. With a right-tail area of 0.05 the critical value is z=1.645. The critical value is used to set up the critcal region. Using formula (6.1), with n=150, $\sigma$=11.3, x=120 and $\mu$=116, the test statistic is 4.34. This value is right of the critical value of 1.645. Hence, the test statistic is in the critical region, which means that the null hypothesis is rejected. Since the null hypothesis is rejected, the alternate hypothesis is supported. Thus, the population mean is greater than 116. The professor is wrong.

Right-Tailed Region

A right-tailed region is the region to the (you guessed it!) right of the critical value.

Two-Tailed Region

A two-tailed region is split into two parts, that are to the right of the critcal value, and to the left of the negative of the critical value.

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