Before the launch of a commercial product, a company makes a market survey to know the price that buyers are willing to pay. It is assumed that this price is normally distributed with a desviation of \$10. The marketing department informs that the public considers appropriate the price of \$30. To test this hypothesis against a price of \$40, a sample of 25 people is selected and adopted the following decision rule: if the sample mean is less than \$35, are considered it is appropriate to set a price of \$30.
- Find the probability of committing an error of type 1.
- Find the probability of committing an error of type 2.
- Find de power.
Let be $\alpha$ the probabilityof committing an error of type 1 and $\beta$ the probability of committing an error of type 2: $\alpha=P($accept $H_0/H_0$ is false$)$ and $\beta=P($refuse $H_0/H_0$ is false$)$
The problem is thatI can't find the hypothesis. My criterion is to define it as follows:
$H_0: \mu = \$30$ -> null hypothesis
$H_1: \mu \not= \$30$ -> alternative hypothesis
This is ok?, but what can I do with the price of $\$40$?
And, whatis the sample mean?, I have to find it to calculate the $\alpha$. Or I have to suppose an $\alpha=0,01$?
let $X$ be a random variable that represents the price.
$H_0: X \sim \mathcal{N}(30, 100) $
$H_1: X \sim \mathcal{N}(40, 100) $
Sample mean is $\bar{X} = \frac{1}{25}\sum\limits_{i=1}^{25}X_i$, where $X_i$ are i.i.d. having the same distribution as $X$.