Question

A software company is trying to decide whether to produce an upgrade of one of its programs. Customers would have to pay $\backslash (100$for the upgrade. For the upgrade to be profitable, the company needs 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 $\backslash )100$for the upgrade.

(a) Do the sample data give good evidence that more than $20\%$of the company’s customers are willing to purchase the upgrade? Carry out an appropriate test at the $A=0.05$significance level.

(b) Which would be a more serious mistake in this setting—a Type I error or a Type II error? Justify your answer.

(c) Other than increasing the sample size, describe one way to increase the power of the test in (a).

Short answer

a). No, There is not sufficient evidence to support the claim.

b). Type I error is worse, because then the upgrade is less likely to be profitable.

c). Increase the significance level.

Step-by-step solution

Part (a) Step 1: Given Information

$x=16$

$n=60$

Part (a) Step 2: Explanation

Determine the hypotheses:

$H_0:p=20\%$

$=0.20$

$H_a:p>0.20$

The sample proportion is the number of successes divided by the sample size:

localid="1650442739498" $$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{16}{60} \end{gathered}$$

$\approx 0.2667$

Determine the value of the test-statistic:

localid="1650442753640" $$\begin{gathered} z=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}} \\ =\frac{0.2667-0.2}{\sqrt{\frac{0.2(1-0.2)}{60}}} \end{gathered}$$

$\approx 1.29$

Part (a) Step 3: Explanation

The $P-$value is the chance of getting the test statistic's result, or a number that is more severe. Calculate the $P-$value in table A as follows:

$P=P(Z>1.29)$

$=P(Z<-1.29)$

$=0.0985$

Reject the null hypothesis if the$P-$value is less than the significance level:

$P>0.05\Rightarrow \text{Fail to reject}{H}_0$

Part (b) Step  1: Given Information

$H_0:p=20\%$

$=0.20$

$H_a:p>0.20$

Part (b) Step 2: Explanation

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

Consequence: Less people are willing to pay $\$100$ than it appears from the results of the test.

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

Consequence: More people are willing to pay $\$100$than it appears from the results of the test.

Type I error is worse, because then the upgrade is less likely to be profitable.

Part (c) Step 1: Given Information

For the upgrade to be profitable, the company needs to sell it to more than $20\%$ of their customers.

Part (c) Step 2: Explanation

You can increase the power by:

Increasing the sample size (because having more information about the population will allow us to make better estimations).

Increase the significance level (because this increases the probability of making a Type I error and decreases the probability of making a Type II error; Since the power is $1$ decreased by the probability of making a Type II error and thus the power increases).

Making the alternative proportion $p$ more extreme (thus $p$ greater than $0.45$, since more extreme alternatives are easier to prove).