## Related questions with answers

A software company is trying to decide whether to produce an upgrade of one of its programs. Customers would have to pay $100 for the upgrade. For the upgrade to be profitable, the company has to sell it to more than 20% of their customers. You contact a random sample of 60 customers and find that 16 would be willing to pay$100 for the upgrade. Which would be a more serious mistake in this setting-a Type I error or a Type II error? Justify your answer.

Solution

VerifiedGiven:

$\begin{align*} n&=\text{Sample size}=60 \end{align*}$

Given claim: more than 20%

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis states that the population proportion is equal to the value mentioned in the claim. If the null hypothesis is the claim, then the alternative hypothesis states the opposite of the null hypothesis.

$\begin{align*} H_0&:p=20\%=0.20 \\ H_a&:p>0.20 \end{align*}$

$\textbf{Type I error}$

Type I error: Reject the null hypothesis $H_0$, when the null hypothesis $H_0$ is true.

There is convincing evidence that more than 20% of the company's customers are willing to purchase the upgrade, when actually 20% of the company's customers are willing to purchase the upgrade.

A possible consequence is that the store will sell the upgrade, while not enough people buy it and thus the store could make a loss.

$\textbf{Type II error}$

Type II error: Fail to reject the null hypothesis $H_0$, when the null hypothesis $H_0$ is false.

There is no convincing evidence that more than 20% of the company's customers are willing to purchase the upgrade, when more than 20% of the company's customers are actually willing to purchase the upgrade.

A possible consequence is that the store does not sell the upgrade, while the store could have made a profit on it.

It is worse for the store to make a loss and thus a type I error is more serious in this case.

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