Respuesta :
Answer:
The 95% confidence interval for the proportion of all boards in this shipment that fall outside the specification is (1.8%, 6.2%).
Step-by-step explanation:
Let X = number of boards that fall outside the most rigid level of industry performance specifications.
In a random sample of 300 boards the number of defective boards was 12.
Compute the sample proportion of defective boards as follows:
[tex]\hat p =\frac{12}{300}=0.04[/tex]
The (1 - α)% confidence interval for population proportion p is:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
The critical value of z for 95% confidence level is,
[tex]z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96[/tex]
*Use a z-table.
Compute the 95% confidence interval for the proportion of all boards in this shipment that fall outside the specification as follows:
[tex]CI=\hat p\pm z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}\\=0.04\pm1.96\sqrt{\frac{0.04(1-0.04)}{300}}\\=0.04\pm0.022\\=(0.018, 0.062)\\\approx(1.8\%, 6.2\%)[/tex]
Thus, the 95% confidence interval for the proportion of all boards in this shipment that fall outside the specification is (1.8%, 6.2%).
