Respuesta :
Answer:
The least number of tennis balls needed for the sample is 1849.
Step-by-step explanation:
The (1 - α) % confidence interval for population proportion is:
[tex]CI=\hat p\pm z_{\alpha/2}\ \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The margin of error for this interval is:
[tex]MOE= z_{\alpha/2}\ \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
Assume that the proportion of all defective tennis balls is p = 0.50.
The information provided is:
MOE = 0.03
Confidence level = 99%
α = 1%
Compute the critical value of z for α = 1% as follows:
[tex]z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex]
*Use a z-table.
Compute the sample size required as follows:
[tex]MOE= z_{\alpha/2}\ \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
[tex]n=[\frac{z_{\alpha/2}\times \sqrt{\hat p(1-\hat p)} }{MOE}]^{2}[/tex]
[tex]=[\frac{2.58\times \sqrt{0.50(1-0.50)} }{0.03}]^{2}\\\\=1849[/tex]
Thus, the least number of tennis balls needed for the sample is 1849.
