9.1 Null and Alternative Hypotheses
A hypothesis test is procedure used to determine whether sample data provides enough evidence to determine the validity of claims made about a population.
A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim[1] about a population parameter.
Null hypothesis is a statement about the value of a population parameter, such as the population mean [latex](\mu)[/latex] or the population proportion [latex](p)[/latex]. It contains the condition of equality and is denoted as H0 (pronounced H-naught). The equality condition indicates the status quo, or no change or no effect or no difference. The null hypothesis is assumed true until sample evidence indicates otherwise.
Examples: [latex]H_0:\mu = 157[/latex] or [latex]H_0:p = 0.35[/latex]
Alternative hypothesis is the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H1 (or Ha).
Examples: [latex]H_a:\mu > 157[/latex] or [latex]H_a:p \ne 0.35[/latex]
If a statement contains the condition of equality ([latex]=[/latex] or [latex]\le[/latex] or [latex]\ge[/latex]), then that statement will be expressed as the null hypothesis. Note that [latex]\le[/latex] indicates less than or equal to, while [latex]\ge[/latex] indicates greater than or equal to. In both cases, the equal to (the condition of equality) part is present.
Understanding the Language of Hypothesis Testing
EXAMPLE 1
A climatologist wants to check the claim on a Wikipedia article that San Diego averages a maximum of 66 days of measurable precipitation per year.
SHOW SOLUTION
Since the question states that “San Diego averages a maximum of 66 days …“, the parameter of interest is the average (or the mean).
Note that for this class we’ll use the words mean and average interchangeably, although mathematically they can be different. Read more about Mean vs Average.
The actual average days of annual precipitation for San Diego is not something the climatologist knows otherwise they could easily see if that average is no more than 66 days. Therefore the unknown parameter here for the researcher is the average number of days of measurable precipitation per year in San Diego. Let’s call that average [latex]\mu[/latex].
Let [latex]\mu =[/latex] average days of measurable precipitation per year in San Diego
Let’s write the claim using mathematical symbols:
the average days of measurable precipitation per year in San Diego [latex]= \mu[/latex]
is no more than 66 [latex]\rightarrow[/latex] [latex]\le 66[/latex]
Putting it together, the claim is: [latex]\mu\le 66[/latex]. This claim contains the condition of equality. The opposite of this claim is [latex]\mu > 66[/latex].
Of the two statements above, the null hypothesis, [latex]H_o[/latex], is the one that has the [latex]=[/latex] sign while other statement is the alternative hypothesis, [latex]H_a \ ([/latex]or [latex]H_1)[/latex]. We can write:
\[H_o: \mu \le 66\]\[H_a: \mu > 66\]
We can represent the null and the alternative hypotheses on a number line. Since the null hypothesis is [latex]\mu\le 66[/latex], the part of the number line representing this inequality is shown by the blue arrow below. This is the side that favors the null. The other unshaded side on the number line represents the alternative hypothesis, which in this case is [latex]\mu > 66[/latex].
Note that the unknown mean, [latex]\mu[/latex], whatever the value is, can not be both on the null hypothesis side indicated by the blue arrow and the unshaded alternative side. This means if one hypothesis is true, the other one must necessarily be false.
Some textbook authors prefer to write the null hypothesis for this example as [latex]H_o:\mu=66[/latex], where 66 is the highest value on the null side. When testing the alternative against the null, if we can reasonably be assured that the unknown parameter is greater than 66, then it’s automatically greater than any value less than 66. With this approach, we write the null and the alternative as follows:
\[H_o: \mu = 66\]\[H_a: \mu > 66\]
EXAMPLE 2
An economist claims that lower than 81% of investment portfolios went up in value this past year.
SHOW SOLUTION
Since the 81% given is the percentage of investment portfolios that went up in value this past year, that 81% represents the proportion of portfolios that increased their value. The economist believes that the percentage of portfolios that increased their value is lower, but they don’t know that actual percentage. The unknown parameter here is the proportion (the decimal equivalent of the percentage) of investment portfolios that went up in value. Let’s call that proportion p.
Let [latex]p =[/latex] proportion of investment portfolios that went up in value this past year
The economist believes that this actual proportion is lower than 81% or 0.81. In symbols:
[latex]p<0.81[/latex]
This is an example of a claim that doesn’t contain the condition of equality. The opposite of the claim is:
[latex]p\ge0.81[/latex]
Of the two statements above, the null hypothesis, [latex]H_o[/latex], is the one that has the [latex]=[/latex] sign while other statement is the alternative hypothesis, [latex]H_a \ ([/latex]or [latex]H_1)[/latex]. We can write:
| [latex]H_0: p \ge 0.81[/latex][latex]H_a: p < 0.81[/latex] | OR | [latex]H_0: p = 0.81[/latex][latex]H_a: p < 0.81[/latex]` |
EXAMPLE 3
The mean price of mid-sized cars in a region is 42400. A test is conducted to see if the claim is true.
SHOW SOLUTION
Since the test is to see if the claim that the mean price of 42400 is true for mid-sized cars, this would be a test for the mean . The unknown parameter here is the mean price of all mid-sized cars. Let’s call that average [latex]\mu[/latex].
Let [latex]\mu =[/latex] mean price of mid-sized cars in a region
We’d like to see if this mean, [latex]\mu[/latex], is equal to 42400. Let’s write the claim about the mean using mathematical symbols:
[latex]\mu = 42400[/latex]
This is an example of a claim that contains condition of equality. The opposite of this claim is [latex]\mu \ne 42400[/latex].
Of the two statements above, the null hypothesis, [latex]H_o[/latex], is the one that has the [latex]=[/latex] sign while other statement is the alternative hypothesis, [latex]H_a \ ([/latex]or [latex]H_1)[/latex]. We can write:
\[H_o: \mu = 42400\]\[H_a: \mu \ne 42400\]
PRACTICE
ADDITIONAL Practice
Test statistic is the z or t-score of the sample data. Some textbooks define test statistic as the sample statistic (mean, proportion, difference of means, etc.) and the z or t-score associated with that sample statistic as the standardized test statistic.
Tail-type of a test is the direction favored by the alternative hypothesis.
Types of Tail
Review Types of Tests.
For a visual explanation, watch the following video:
How do you decide the tail type for your study?
Choosing between One-tailed or a two-tailed test for your data analysis
