Respuesta :
Answer:
(a) The 90 percent confidence interval for the population mean yearly premium is ($10,974.53, $10983.47).
(b) The sample size required is 107.
Step-by-step explanation:
(a)
The (1 - α)% confidence interval for population mean is:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
Given:
[tex]\bar x=\$10,979\\s=\$1000\\n=20[/tex]
Compute the critical value of t for 90% confidence level as follows:
[tex]t_{\alpha/2, (n-1)}=t_{0.10/2, (20-1)}=t_{0.05, 19}=1.729[/tex]
*Use a t-table.
Compute the 90% confidence interval for population mean as follows:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
[tex]=10979\pm 1.729\times \frac{1000}{\sqrt{20}}\\=10979\pm4.47\\ =(10974.53, 10983.47)[/tex]
Thus, the 90 percent confidence interval for the population mean yearly premium is ($10,974.53, $10983.47).
(b)
The margin of error is provided as:
MOE = $250
The confidence level is, 99%.
The critical value of z for 99% confidence level is:
[tex]z_{\alpha/2}=z_{0.01/2}=z_{0.005}=2.58[/tex]
Compute the sample size as follows:
[tex]MOE= z_{\alpha/2}\times \frac{s}{\sqrt{n}}[/tex]
[tex]n=[\frac{z_{\alpha/2}\times s}{MOE} ]^{2}[/tex]
[tex]=[\frac{2.58\times 1000}{250}]^{2}[/tex]
[tex]=106.5024\\\approx107[/tex]
Thus, the sample size required is 107.
