8.1 Hypotheses
The first step of a hypothesis test is the formulation of the hypotheses. A hypothesis is a statement about the value of a parameter. For example, [latex]\mu \neq 100[/latex] ml, [latex]\mu <100[/latex] ml or [latex]\mu >100[/latex] ml, etc.
There are two hypotheses in a hypothesis test: the null hypothesis,denoted as [latex]H_0[/latex]; and the alternative hypothesis, denoted as [latex]H_a[/latex]. The alternative hypothesis is a claim we wish to establish regarding a population parameter, and the null hypothesis is the opposite claim. More specifically, the alternative hypothesis represents the plausible values of the parameter if we reject the null hypothesis. Therefore, the null and alternative hypotheses must be complimentary statements about the population parameter.
There are two possible outcomes in a hypothesis test: reject the null [latex]H_0[/latex] in favour of the alternative [latex]H_a[/latex] or do not reject [latex]H_0[/latex]. Rejecting a hypothesis means that the hypothesis is claimed to be false, and accepting a hypothesis implies that the hypothesis is claimed to be true. If sample data provide strong evidence against the null hypothesis, it is rejected in favor of the alternative hypothesis; if the data do not provide strong evidence against the null hypothesis, we fail to reject it. Failure to reject the null hypothesis does not imply the null hypothesis is true, but rather, it could be true.
For example, the legal systems of many countries act in accordance with the presumption of innocence, under which an individual is presumed innocent until proven guilty. In such countries, when an individual has been charged with a crime, the null and alternative hypotheses are [latex]H_0[/latex] the defendant is innocent versus [latex]H_a[/latex] the defendant is guilty. The defendant will be found guilty ([latex]H_0[/latex] rejected in favor of [latex]H_a[/latex]) only if there is strong evidence to indicate their guilt; if the evidence provided is not sufficient, the defendant is found not guilty ([latex]H_0[/latex] is not rejected). However, in the case where the defendant is found not guilty, it is not because there is strong evidence to establish their innocence but rather insufficient evidence to prove their guilt.
Steps to set up the hypotheses:
- Look for the keywords and write down what we want to claim under the alternative [latex]H_a[/latex].
- Write the opposite of the alternative [latex]H_a[/latex] to obtain the null [latex]H_0[/latex].
Depending on the purpose of the hypothesis test, there are three choices for [latex]H_a[/latex]:
Table 8.1: Key Words and Choices of Alternative Hypothesis
|
Two tailed |
Right (upper) tailed |
Left (lower) tailed |
|
[latex]H_a: \mu \neq \mu_0[/latex] |
[latex]H_a: \mu \: > \: \mu_0[/latex] |
[latex]H_a: \mu < \mu_0[/latex] |
|
“differ”, “change” |
“more than”, “increase” |
“less than”, “decrease” |
Note: The notation [latex]\mu_0[/latex] represents a specific hypothesized value for the population mean [latex]\mu[/latex]. It will be replaced by a certain number in context. For example, [latex]\mu_0=7[/latex] in the following example.
Example: Set Up Hypotheses
Newborn babies in 2010 had an average weight of 7 pounds. Write the hypotheses to test whether:
- The average weight in 2015 has changed.
Steps:- Key word: “changed”→ a two-tailed test. Write down the alternative: [latex]H_a: \mu \neq 7 \text{ lb}[/latex]
- Write the opposite of the alternative to obtain the null [latex]H_0: \mu = 7 \text{ lb}[/latex]
Therefore, the hypotheses are [latex]H_0: \mu = 7 \text{ lb}[/latex] versus [latex]H_a: \mu \neq 7 \text{ lb}[/latex].
- The average weight in 2015 has increased.
Steps:- Key word: “increased” → a right-tailed test. Write down the alternative: [latex]H_a: \mu > 7 \text{ lb}[/latex].
- Write the opposite of the alternative to obtain the null [latex]H_0: \mu \leq 7 \text{ lb}[/latex].
Therefore, the hypotheses are [latex]H_0: \mu \leq 7 \text{ lb}[/latex] versus [latex]H_a: \mu > 7 \text{ lb}[/latex]
- The average weight in 2015 has decreased.
Steps:- Key word: “decreased” → a left-tailed test. Write down the alternative: [latex]H_a: \mu < 7 \text{ lb}[/latex]
- Write the opposite of the alternative to obtain the null [latex]H_0: \mu \geq 7 \text{ lb}[/latex]
Therefore, the hypotheses are [latex]H_0: \mu \geq 7 \text{ lb}[/latex] versus [latex]H_a: \mu < 7 \text{ lb}[/latex].
Exercises: Set up Hypotheses for the Following Studies
- A machine fills beer into bottles whose volume is supposed to be 341 ml, but the exact amount varies from bottle to bottle. We randomly picked 100 bottles and obtained the sample mean volume of 339 ml. Assume the population standard deviation [latex]\sigma=5[/latex] ml. Test at the 5% significance level whether the machine is NOT working properly.
- A computer company claims that its laptops have an average lifetime of about 4 years. A simple random sample of 36 laptops yields an average lifetime of 3.5 years with a sample standard deviation of 4.2 years. Test at the 1% significance level whether this brand of laptops has a mean lifetime of less than 4 years.
- The number of cell phone users has increased dramatically since 1997. Suppose the mean local monthly bill was $50 for cell phone users in the United States in 2006. A simple random sample of 50 cell phone users was obtained in 2019, and the sample mean local monthly bill was [latex]\bar x=55[/latex] with a sample standard deviation [latex]s[/latex] = $25. At the 10% significance level, do the data provide sufficient evidence to conclude that the 2019 mean local monthly bill for cell phone users has increased from the 2006 mean of $50?
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Answers:
- [latex]H_0: \mu = 341[/latex] ml versus [latex]H_a: \mu \neq 341[/latex] ml
- [latex]H_0: \mu \geq 4[/latex] years versus [latex]H_a: \mu \lt 4[/latex] years
- [latex]H_0: \mu \leq $50[/latex] versus [latex]H_a: \mu \gt $50[/latex]
