Question

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 needs to sell it to more than 25% of its customers. You contact a random sample of 72 customers and find that 19 would be willing to pay $100 for the upgrade.
a) Do the sample data give good evidence that more than 25% of the company’s customers are willing to purchase the upgrade? Carry out an appropriate test at the α = 0.05 significance level. (You must include all of the steps of the STATE, PLAN, DO, CONCLUDE method to earn full credit.)

b) Describe in context the consequence if we commit a Type II error in this case.

Answer 1

A)  The critical value at α = 0.05  will be 1.645 since the critical value does not fall within the rejection region we reject the null hypothesis

B) A type II error occurs when we fail to reject the null hypothesis and a type II error is also known as an error of omission and the consequence of this type of error is that a false negative result is produced.

Given data :

Amount paid for upgrade = $100

Percentage of customers needed  = 25% = 0.25

Sample size ( n ) = 72

Number willing to pay $100 = 19

significance level ( α ) = 0.05

A) Prove that 25% of the company's customers are willing to purchase the upgrade

Null hypothesis ( H0 ) : p = 0.25

Alternate hypothesis ( H₁ ) : p > 0.25

α  = 0.05

Ρ = 19 / 72 = 0.26

To determine the hypothesis to reject or accept

perform Test statistic :  Z = $\frac{P - p }{\sqrt{\frac{p(1-p)}{n} } }$  

                                          = ( 0.26 - 0.25 ) / $\sqrt{0.25 * 0.75} / 72$

                                          = 0.01 / 0.0026

                                      Z- score = 0.499

The critical value at α = 0.05  will be 1.645 since the critical value does not fall within the rejection region we reject the null hypothesis

B)  A type II error occurs when we fail to reject the null hypothesis and a type II error is also known as an error of omission and the consequence of this type of error is that a false negative result is produced.

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