Respuesta :
Answer:
(a) The sample size required is 43.
(b) The sample size required is 62.
Step-by-step explanation:
The (1 - α) % confidence interval for population mean is:
[tex]CI=\bar x\pm z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}[/tex]
The margin of error for this interval is:
[tex]MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}[/tex]
(a)
The information provided is:
σ = 4 minutes
MOE = 72 seconds = 1.2 minutes
Confidence level = 95%
α = 5%
Compute the critical value of z for α = 5% as follows:
[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use a z-table.
Compute the sample size required as follows:
[tex]MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}[/tex]
[tex]n=[\frac{z_{\alpha/2}\times \sigma}{MOE} ]^{2}[/tex]
[tex]=[\frac{1.96\times 4}{1.2}]^{2}\\\\=42.684\\\\\approx 43[/tex]
Thus, the sample size required is 43.
(b)
The information provided is:
σ = 4 minutes
MOE = 1 minute
Confidence level = 95%
α = 5%
Compute the critical value of z for α = 5% as follows:
[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use a z-table.
Compute the sample size required as follows:
[tex]MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}[/tex]
[tex]n=[\frac{z_{\alpha/2}\times \sigma}{MOE} ]^{2}[/tex]
[tex]=[\frac{1.96\times 4}{1}]^{2}\\\\=61.4656\\\\\approx 62[/tex]
Thus, the sample size required is 62.
