Q.11 - The Practice of Statistics for AP

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). 

Step-by-Step Solution

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.

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:

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

   $\approx 0.2667$

Determine the value of the test-statistic:

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

 $\approx 1.29$

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).