Answer to Question #257427 in Statistics and Probability for miel

Question

Problem. A certain bank issues its own credit card. The head of credit collection  department found out that the mean amount of unpaid balance of the credit card holders was Php 25,000.00 and with a standard deviation of  Php 1,500.00. The head of the bank wanted to find out whether the mean amount of unpaid balance was really Php 25,000.00. He sample 40 credit card holders and found out a sample mean of Php 23,900.00. Can the head of the credit collection department conclude that the unpaid balance is less than Php 25,000.00? Use the 0.01 alpha level. Illustrate the data using a Normal Distribution Curve. Refer the critical value from the Z-Scores Distribution Table or Standard Normal Probabilities Table. 

Answer 1

The hypotheses tested in this question are,

$H_0:\mu=25000\space vs\space H_1:\mu\lt 25000$

We are given that,

$\bar{x}=23900,\space n=40,\space \sigma=1500$

Since the population standard deviation is known and the sample size is greater than 30, we will use the standard normal distribution to perform this test as shown below.

The test statistic is given as,

$Z_c=(\bar{x}-\mu)/(\sigma/\sqrt{n})$

Substituting for the values we have,

$Z_c=(23900-25000)/(1500/\sqrt{40})$

$Z_c=(-1100)/237.1708=-4.683(3dp)$

$Z_c$ is compared with the standard normal table value at $\alpha=0.01$. The table value is given as,

$Z_{\alpha}=Z_{0.01}=2.33$.

The null hypothesis is rejected if $Z_c\lt -Z_{0.01}$

Since $Z_c=-4.683\lt -Z_{0.01}=-2.33,$ we reject the null hypothesis and hence there is sufficient evidence for the head of the credit collection department to conclude that that the unpaid balance is less than Php 25,000.00 at $\alpha=0.01$ level of significance.

The Normal Distribution Curve is given below.