9.1 Null and Alternative Hypotheses

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 $\mu$.

Let $\mu =$ 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 $=\mu$

is no more than 66  $\to$      $\le 66$

Putting it together, the claim is: $\mu \le 66$. This claim contains the condition of equality. The opposite of this claim is $\mu >66$.

Of the two statements above, the null hypothesis, $H_o$, is the one that has the $=$ sign while other statement is the alternative hypothesis, $H_a($or $H_1)$. We can write:

$$H_o:\mu \le 66$$$$H_a:\mu >66$$

Some textbook authors prefer to write the null hypothesis for this example as $H_o:\mu =66$, 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:

$$H_o:\mu =66$$$$H_a:\mu >66$$

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 $p=$ 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:

$p<0.81$

This is an example of a claim that doesn’t contain the condition of equality. The opposite of the claim is:

$p\ge 0.81$

Of the two statements above, the null hypothesis, $H_o$, is the one that has the $=$ sign while other statement is the alternative hypothesis, $H_a($or $H_1)$. We can write:

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 $\mu$.

Let $\mu =$ mean price of mid-sized cars in a region

We’d like to see if this mean, $\mu$, is equal to 42400. Let’s write the claim about the mean using mathematical symbols:

$\mu =42400$

This is an example of a claim that contains condition of equality. The opposite of this claim is $\mu \ne 42400$.

Of the two statements above, the null hypothesis, $H_o$, is the one that has the $=$ sign while other statement is the alternative hypothesis, $H_a($or $H_1)$. We can write:

$$H_o:\mu =42400$$$$H_a:\mu \ne 42400$$