Answer: 471
Step-by-step explanation:
Given : The population standard deviation is estimated to be $500.
i.e . [tex]\sigma=\$500[/tex]
If a 97 percent confidence interval is used and an interval of $100 is desired.
i.e. Margin of error = half of interval
i.e. E= [tex]\dfrac{\$100}{2}=\$50[/tex]
Significance level : 1-0.97=0.03
Critical value for 97% confidence interval : [tex]{z_{\alpha/2}=z_{0.03/2}=2.17[/tex]
Formula for sample size :
[tex]n=(\dfrac{z_{\alpha/2}\ \sigma}{E})^2\\\\=(\dfrac{2.17\times500}{50})^2\\\\=470.89\approx471[/tex]
Hence, at-least 471 cardholders should be sampled.
