Chapter 9, Problem 11PT

Solution

a)

To verify the claim that there are more than 20% of the company’s customers are willing to purchase the upgrade.

The sample data does not give good evidence that more than 20% of the company’s customers are willing to purchase the upgrade.

Given:Claim is customers would have pay $100 for the upgrade and company need to sell it to more than 20% of their customers to be profitable.

Sample size = 60

Number of people willing to pay $100 for the upgrade = 16

Concept used:The conditions to satisfy to run one sample proportion Z test

1) The sample should be random

2) np = 10

3) n(1-p) = 10

4) There should be at most 10% of people of population in the sample.

Formula used:$Z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$

Calculation:

Let p be proportion of those people who are ready to pay $100 for upgrade. The null and alternate hypothesis are shown below

  $\begin{array}{l}{H}_0:p=0.20\\ H_a:p>0.20\end{array}$

The sample of 60 customers is drawn randomly.

The sample size is 60, there will be at least 600 customers in total. So the 10% condition is also met.

The proportion of customers in sample who are willing to pay $100 for upgrade

  $\hat{p}=\frac{16}{60}=0.27$

  $\begin{array}{l}n\hat{p}=60*0.27=16\\ n\hat{p}>10\\ n(1-\hat{p})=60*(1-0.27)=44\\ n(1-\hat{p})>10\end{array}$

All conditions are met so we can use one sample Z proportion test.

The test statistic is calculated as shown below

  $\begin{array}{l}Z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}=\frac{0.27-0.20}{\sqrt{\frac{0.20*(1-0.20)}{60}}}\\ Z=1.29\end{array}$This is one tail test and the p value is P(Z = 1.29), with respect to table we get p value is 0.0984.

The p value is greater than the level of significance 0.05. So at 5% level of significance there is no evidence to reject the null hypothesis. So, there is no evidence to conclude that the proportion of customers ready to pay $100 for new upgrade is more than 20%.

b)

The Type-I and Type-II error in the given setting and serious mistake in this setting.

Given:A significance test is performed to test the claim that the proportion of customers who are willing to pay $100 for new upgrade is more than 20%.The p value of a significance test as 0.0984.

Concept used:Type-I error : Rejecting the true null hypothesis or it is also known as false positive.

Type-II error: Non rejection of a false null hypothesis or it is also known as false negative.

Explanation:Type-I error : Concluding that the proportion of customers who are willing to pay $100 for new upgrade is more than 20% but actually it is not true.

Type-II error: Concluding that the proportion of customers who are willing to pay $100 for upgrade is not more than 20% but actually it is not true.

If we commit Type-I error and conclude that the proportion of customers who are willing to pay $100 for new upgrade is more than 20% then it will create more serious problem as company would not be profitable as the proportion of customers willing to pay $100 is not actually more than 20%.

c)

One more way to increase the power of the hypothesis test other than increasing sample size.

Explanation:For maximizing the power of test, either we have to increase the sample size or increase the significance level.

The other way to in increase power of the test is to increase the significance .