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:
![CI=\bar x\pm z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=CI%3D%5Cbar%20x%5Cpm%20z_%7B%5Calpha%2F2%7D%5C%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
The margin of error for this interval is:
![MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5C%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
(a)
The information provided is:
<em>σ</em> = 4 minutes
MOE = 72 seconds = 1.2 minutes
Confidence level = 95%
α = 5%
Compute the critical value of z for α = 5% as follows:
![z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.05%2F2%7D%3Dz_%7B0.025%7D%3D1.96)
*Use a z-table.
Compute the sample size required as follows:
![MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5C%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
![n=[\frac{z_{\alpha/2}\times \sigma}{MOE} ]^{2}](https://tex.z-dn.net/?f=n%3D%5B%5Cfrac%7Bz_%7B%5Calpha%2F2%7D%5Ctimes%20%5Csigma%7D%7BMOE%7D%20%5D%5E%7B2%7D)
![=[\frac{1.96\times 4}{1.2}]^{2}\\\\=42.684\\\\\approx 43](https://tex.z-dn.net/?f=%3D%5B%5Cfrac%7B1.96%5Ctimes%204%7D%7B1.2%7D%5D%5E%7B2%7D%5C%5C%5C%5C%3D42.684%5C%5C%5C%5C%5Capprox%2043)
Thus, the sample size required is 43.
(b)
The information provided is:
<em>σ</em> = 4 minutes
MOE = 1 minute
Confidence level = 95%
α = 5%
Compute the critical value of z for α = 5% as follows:
![z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.05%2F2%7D%3Dz_%7B0.025%7D%3D1.96)
*Use a z-table.
Compute the sample size required as follows:
![MOE=z_{\alpha/2}\ \frac{\sigma}{\sqrt{n}}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5C%20%5Cfrac%7B%5Csigma%7D%7B%5Csqrt%7Bn%7D%7D)
![n=[\frac{z_{\alpha/2}\times \sigma}{MOE} ]^{2}](https://tex.z-dn.net/?f=n%3D%5B%5Cfrac%7Bz_%7B%5Calpha%2F2%7D%5Ctimes%20%5Csigma%7D%7BMOE%7D%20%5D%5E%7B2%7D)
![=[\frac{1.96\times 4}{1}]^{2}\\\\=61.4656\\\\\approx 62](https://tex.z-dn.net/?f=%3D%5B%5Cfrac%7B1.96%5Ctimes%204%7D%7B1%7D%5D%5E%7B2%7D%5C%5C%5C%5C%3D61.4656%5C%5C%5C%5C%5Capprox%2062)
Thus, the sample size required is 62.