Answer to Question #137828 in Statistics and Probability for Maxwell

Question

ATM machines have made it easier and faster for customers to check account balances, transfer money, and obtain cash. Many banks are now considering the next generation of ATM machines, which will feature news headlines, full-motion video, and tickets to events. In a random sample of 500 customers, 280 said ATM machines should offer postage stamps. (20 marks)
a. Find the sample proportion of customers who believe ATM machines should offer postage stamps.
b. Find a 99% confidence interval for the true proportion of customers who believe ATM machines should offer postage stamps.
c. A bank official claims the proportion of customers who believe ATM machines should offer postage stamps is 0.60. Is there any evidence to refute this claim? Test at the 5% level
d. If you test at the 1% level, what would be your decision?

Answer 1

solution

Part a)

$p = \frac{280}{500}= \frac{14}{25}=0.56$

answer: 0.56

part b)

At a $99\%$ level of confidence, the test statistic $Z= 2.576$

Standard deviation $\sigma = \sqrt \frac{p(1-p)}{n}$

$\sqrt \frac{0.56(1-0.56)}{500}=0.0222$

The confidence interval:

$$p-z*\sigma < p < p+z*\sigma$$ $$0.56-2.576*0.0222 < p < 0.56+2.576*0.0222$$ $$0.5028<p<0.6172$$

answer: $[50.28\%, 61.72\%]$

Part c)

$\hat p=0.6$

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

$$\frac{0.56-0.6}{\sqrt \frac{0.6(1- 0.6)}{500} }=-1.876$$

At a 95% level of confidence, the rejection region is $|Z|>1.96$

answer: since $|-1.876| \not >1.96$ we fail to reject the hypothesis. There is no evidence to refute the claim

Part d)

From part b), we fail to reject the claim if it is within the confidence limits i.e.

$[50.28\%, 61.72\%]$

Answer: We fail to reject the claim since 0.6 is within the acceptance region