A supplier has stated that the proportion of defective items in a very large batch of components is 5%. The customer has decided to check this statement by examining every item in a sample of 300 items selected at random from the batch. The sample was found to contain 21 defective items and 279 perfect items. Determine the lower limit, if a 95% confidence interval is set up for the proportion of all defective items.

Answer 1

To determine the lower limit of the confidence interval for the proportion of all defective items, we can use the following formula:

Lower limit = sample proportion - margin of error

To calculate the sample proportion, we divide the number of defective items in the sample by the total sample size:

Sample proportion = defective items / total sample size
= 21 / 300
= 0.07

The margin of error can be calculated using the following formula:

Margin of error = critical value * standard error

First, we need to determine the critical value for a 95% confidence interval. Since we are looking for the lower limit, we will use a one-tailed test. The critical value for a one-tailed 95% confidence interval is approximately 1.645.

Next, we need to calculate the standard error, which is the standard deviation of the sampling distribution. The formula for the standard error of a proportion is:

Standard error = sqrt[(sample proportion * (1 - sample proportion)) / total sample size]

Standard error = sqrt[(0.07 * (1 - 0.07)) / 300]
= sqrt[0.000203333]
= 0.014259

Finally, we can calculate the margin of error:

Margin of error = 1.645 * 0.014259
= 0.023463

Now we can calculate the lower limit:

Lower limit = sample proportion - margin of error
= 0.07 - 0.023463
= 0.046537

Therefore, the lower limit of the 95% confidence interval for the proportion of all defective items is approximately 0.0465 (or 4.65%).