8.1 Null and Alternative Hypotheses

A hypothesis test begins by considering two hypotheses. They are called the null hypothesis and the alternative hypothesis. These hypotheses contain opposing viewpoints and only one of these hypotheses is true. The hypothesis test determines which hypothesis is most likely true.

• The null hypothesis is denoted [latex]H_0[/latex]. It is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
  • The null hypothesis is a claim that a population parameter equals some value. For example, [latex]H_0:\mu=5[/latex].
• The alternative hypothesis is denoted [latex]H_a[/latex]. It is a claim about the population that is contradictory to the null hypothesis and is what we conclude is true if we reject [latex]H_0[/latex].
  • The alternative hypothesis is a claim that a population parameter is greater than, less than, or not equal to some value. For example, [latex]H_a:\mu>5[/latex], [latex]H_a:\mu<5[/latex], or [latex]H_a:\mu\neq 5[/latex].  The form of the alternative hypothesis depends on the wording of the hypothesis test.
  • An alternative notation for [latex]H_a[/latex] is [latex]H_1[/latex].

Because the null and alternative hypotheses are contradictory, we must examine the evidence to decide if we have enough evidence to reject the null hypothesis or not reject the null hypothesis. In statistics, the evidence is in the form of sample data. The sample data will either support the claim that the null hypothesis is true or will be strong enough to support the claim of the alternative hypothesis. After we have determined which hypothesis the sample data supports, we make a decision about the validity of the null hypothesis. There are two options for the decision. They are “reject [latex]H_0[/latex]” if the sample information favours the alternative hypothesis or “do not reject [latex]H_0[/latex]” if the sample information is insufficient to reject the null hypothesis.

Exercises

1. You are testing that the mean speed of your cable Internet connection is more than three Megabits per second. State the null and alternative hypotheses.
   
   Click to see Answer

   [latex]\begin{eqnarray*}H_0:&&\mu=3\text{ megabits per second}\\\\H_a:&&\mu\gt 3\text{ megabits per second}\end{eqnarray*}[/latex]

    

2. The mean entry-level salary of an employee at a company is [latex]\$58,000[/latex]. You believe it is higher for IT professionals in the company. State the null and alternative hypotheses.
   
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   [latex]\begin{eqnarray*}H_0:&&\mu=\$58,000\\\\H_a:&&\mu\gt\$58,000\end{eqnarray*}[/latex]

    

3. A sociologist claims the probability that a person picked at random in Times Square in New York City is visiting the area is [latex]83\%[/latex]. You want to test to see if the claim is correct. State the null and alternative hypotheses.
   
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   [latex]\begin{eqnarray*}H_0:&&p=83\%\\\\H_a:&&p\neq 83\%\end{eqnarray*}[/latex]

    

4. In a population of fish, approximately [latex]42\%[/latex] are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses.
   
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   [latex]\begin{eqnarray*}H_0:&&p=42\%\\\\H_a:&&p\lt 42\%\end{eqnarray*}[/latex]

    

5. Suppose that a recent article stated that the mean time spent in jail by a first–time convicted burglar is [latex]2.5[/latex] years. A study was then done to see if the mean time has increased in the new century. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be?
   
   Click to see Answer

   [latex]\begin{eqnarray*}H_0:&&\mu=2.5\text{ years}\\\\H_a:&&\mu\gt 2.5\text{ years}\end{eqnarray*}[/latex]

    

6. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be [latex]15[/latex] years, what would the null and alternative hypotheses be?
   
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   [latex]\begin{eqnarray*}H_0:&&\mu=15\text{ years}\\\\H_a:&&\mu\gt 15\text{ years}\end{eqnarray*}[/latex]

    

7. The National Institute of Mental Health published an article stating that in any one-year period, approximately [latex]9.5\%[/latex] of American adults suffer from depression or a depressive illness. If you were conducting a hypothesis test to determine if the true proportion of people in that town suffering from depression or a depressive illness is lower than the percent in the general adult American population, what would the null and alternative hypotheses be?
   
   Click to see Answer

   [latex]\begin{eqnarray*}H_0:&&p=9.5\%\\\\H_a:&&p\lt 9.5\%\end{eqnarray*}[/latex]

    

8. Previously, an organization reported that teenagers spent [latex]4.5[/latex] hours per week, on average, on the phone. The organization thinks that, currently, the mean is higher. State the null and alternative hypotheses.
   
   Click to see Answer

   [latex]\begin{eqnarray*}H_0:&&\mu=4.5\text{ hours per week}\\\\H_a:&&\mu\gt 4.5\text{ hours per week}\end{eqnarray*}[/latex]

    

“8.2 Null and Alternative Hypotheses” and “8.9 Exercises” from Introduction to Statistics by Valerie Watts is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.