Respuesta :
Answer:
(a) 167
(b) 7
(c) 97
Step-by-step explanation:
The (1 - α)% confidence interval for the population mean μ is:
[tex]CI=\bar x\pm z_{\alpha/2}\cdot\frac{\sigma}{\sqrt{n}}[/tex]
The margin of error is:
[tex]MOE=z_{\alpha/2}\cdot\frac{\sigma}{\sqrt{n}}[/tex]
Then the formula to estimate the sample size is:
[tex]n=[\frac{z_{\alpha/2}\cdot \sigma }{MOE}]^{2}[/tex]
(a)
For 99% confidence interval the critical value of z is:
z = 2.58.
The standard deviation is, 250.
Compute the sample size as follows:
[tex]n=[\frac{z_{\alpha/2}\cdot \sigma }{MOE}]^{2}[/tex]
[tex]=[\frac{2.58\times 250}{50}]^{2}\\\\=(12.9)^{2}\\\\=166.41\\\\\approx 167[/tex]
The sample size that should be used is 167.
(b)
Now the standard deviation is, 50.
Compute the sample size as follows:
[tex]n=[\frac{z_{\alpha/2}\cdot \sigma }{MOE}]^{2}[/tex]
[tex]=[\frac{2.58\times 50}{50}]^{2}\\\\=(2.58)^{2}\\\\=6.6564\\\\\approx 7[/tex]
The sample size that should be used is 7.
(c)
Now a 95% confidence level is used.
For 95% confidence interval the critical value of z is:
z = 1.96.
Compute the sample size as follows:
[tex]n=[\frac{z_{\alpha/2}\cdot \sigma }{MOE}]^{2}[/tex]
[tex]=[\frac{1.96\times 250}{50}]^{2}\\\\=(9.8)^{2}\\\\=96.04\\\\\approx 97[/tex]
The sample size that should be used is 97.
