the percent defective for parts produced by a manufacturing process is targeted at 4%. The process is monitored daily by taking samples of sizes n=160 units. Suppose that today's sample contains 14 defectives. How many units would have to be sampled to be 95% confident that you can estimate the fraction of defective parts within 2% (using the information from today's sample-that is using the result that P=0.00875)?

Formula to find sample size:

n = [(z-value)^2 * p * q]/E^2
... where n = sample size, z-value is found using a z-table for 95% confidence (which is 1.96), p = .0875, q = 1 - p, ^2 means squared, * means to multiply, and E = .02.

Plug values into the formula and calculate n. Round the answer to the next highest whole number.

After calculating the sample size needed to estimate a population proportion to within 0.05, you have been told that the maximum allowable error (E) must be reduced to just 0.025. If the original calculation led to a sample size of 1000, the sample size will now have to be .

4000

To determine the sample size required to estimate the fraction of defective parts within a specified margin of error, we can use the formula:

n = (Z^2 * p * (1-p)) / E^2

Where:
n = required sample size
Z = Z-value corresponding to the desired level of confidence (95% in this case)
p = estimated proportion of defective parts (0.00875, calculated as 14 defectives out of 160 units)
E = desired margin of error (0.02)

First, let's find the Z-value for a 95% confidence level. The Z-value can be obtained from a standard normal distribution table or calculated using statistical software. In this case, we'll use a Z-value of 1.96 (which corresponds to a 95% confidence level).

Plugging the values into the formula:

n = (1.96^2 * 0.00875 * (1 - 0.00875)) / 0.02^2

Simplifying the equation:

n = (3.8416 * 0.00875 * 0.99125) / 0.0004
n = 0.0336 / 0.0004
n = 84

Therefore, in order to be 95% confident that we can estimate the fraction of defective parts within a 2% margin of error, we would need to sample at least 84 units.